What follows is an objective and an "ecumenical" answer from BOTH the (Reformed) Baptist tradition and the Catholic tradition. This is ALSO not a pastoral answer, but objective information that you can use to decide for yourself whether to return to the Catholic church or to stay in the Baptist church.
I now understand God and Jesus. I have fully accepted them into my life and heart. I want to do the believers baptism at that church to let every one know that I now have God in my life and heart. Am I allowed? ... I will always be a Catholic in my heart but I consider myself "born again Catholic".
In the Catholic tradition, God gave you sanctifying grace IN the sacrament of baptism, which was given to you when you were baptized as an infant. By this grace, the Trinitarian life started in you (CCC 2021-2023), which is synonymous with being "born again". Also, because you were baptized IN the Catholic church, you have become a Catholic since then. So from the Catholic perspective, saying "born again Catholic" is redundant, since being "born again" is included in being "Catholic". It is also a misnomer if you bring the "born again" meaning from Protestantism, confusing the two traditions. In Catholicism "faith" is a light given to you as a virtue (part of the sanctifying grace) by which you can assent (later when you reached the age of reason) to the statements about God and Jesus (in the creed) and to the dogmas the Church proposes for you to believe. But this light can be clouded or squandered by mortal sin rendering it ineffective, which becomes evident for Catholics who later become apostate. Thus, if until now have NOT rejected the creed, you already have faith, despite still needing the faith to grow more into a mature understanding (hence the motto 'faith seeking understanding'). It seems that the Baptist church has helped you understand more of your faith in Christ, by teaching you the Bible from which you know more of God and Jesus, which should have been covered in your first communion catechism using a book such as this Saint Joseph Baltimore Catechism.
In the Baptist tradition, "regeneration" (reception of the Trinitarian life, "born again") happens unconsciously for the elect (prior to baptism), which in turn is the prerequisite of "faith" defined as a conscious assent to the gospel properly understood (unlike the Catholic definition of faith as virtue or faith as light), which in turn prompted a person's decision to want to be baptized as a public declaration (as a SEAL of the regeneration that happened earlier). Baptists emphasize understanding the significance of accepting Christ as a prerequisite to (believer) baptism. That is why they don't accept your infant baptism because a baby couldn't understand the significance of being born again and the tremendous gift of salvation through Jesus Christ.
Fortunately, both Evangelical Protestants (which includes Baptists) and Catholics have done much dialogue since Vatican II and have removed much animosities and misunderstanding, recognizing that all groups worship the same Jesus and the same God, despite having irreconcilable beliefs about Mary, the Pope, the nature of the Catholic Sacraments, the Priesthood, and a few other minor differences. The commonality is a lot BIGGER than the differences, as both are rooted in the same catholic tradition (with a lower case "c") as we confess the same ancient creeds (Nicene and Apostle's creeds).
Despite the commonality, speaking as someone who has been studying Catholicism, and in the process I came to understand better my own Protestant upbringing, I wouldn't recommend keeping both identities within yourself: either be a 100% Catholic and remedy what was missing in your Catholic life or be a 100% Protestant (explained below).
What follows is a fuller answer on what is essential to consider should you choose to continue being Catholic, or to transition to becoming a Protestant.
As explained above, your infant baptism was a non-event so your impending believer baptism is NOT "re-baptism". Now that you fully understand that you want to die with Christ by immersion in water and to rise with Christ to participate in His resurrection they deem you to be ready for believer baptism. Baptists are very explicit in saying that Baptism is an ordinance, not a sacrament. Instead, believer baptism is a lot closer to marriage, in that it's a public declaration of a promise already made (with full understanding).
Because it's not a sacrament, baptism does NOT infuse the Trinitarian life to you. The fact that you now understand the significance of Jesus being your Lord and Savior and that you WANT to be baptized, shows that regeneration ("born again") has already happened. In other words, unlike in Catholicism, we don't quite know WHEN regeneration happens; we just know it already happened. Believer baptism in the Baptist tradition then, acts as a SEAL of what already happened, not the MEANS to obtain the Trinitarian life. Not even faith is the means, but grace. Faith makes explicit what God already did by grace (giving us the Trinitarian life). Grace enables faith which then enables our conscious conviction that God has already adopted us into his family through election.
While they don't necessarily deny that you have faith in Jesus prior to going to the Baptist church (despite that you only become fully aware of the significance of being "born again" NOW), they still want you to make it public, and your intention to do the believer baptism makes sense. Public baptism in the Baptist tradition is a truly joyful event (as in weddings) and you get the sense that the congregation is celebrating with you, and thus you have tangible sense of being part of the people of God (I recently witnessed my teenage kid baptized in a similar tradition).
Some less stringent Baptist churches would probably not require you to do believer's baptism, leaving it to your choice, especially if you have "outed" yourself in a public way by some other means. In my case, I was baptized as an infant and around age 15 after catechism, I went through the Reformed Church confirmation ceremony, which was a public event. So several Baptist churches I attended in the past didn't require me to be re-baptized.
As for your Catholic identity, you will lose more and more of your Catholic distinctiveness because as you continue practicing Christianity as a Baptist those Catholic-specific practices become irrelevant. So one point to consider is whether you don't mind losing them, such as understanding the Eucharist as Transubstantiation (along with the practice of Eucharistic adoration), going to confession to receive explicit absolution from a priest, praying to Mary and to the Saints, etc. All those practices will be redefined solely as spiritual relationship to Christ.
There are at least 4 key points to consider.
First, it's your status. From the Catholic church perspective, once a Catholic you are always a Catholic. If you stop going to mass they will categorize you a Lapsed Catholic, a Catholic who is non-practicing. But missing Sunday Mass obligation without a just reason is considered a mortal sin.
Secondly, it's about the Eucharist. Receiving the Eucharist in a Protestant church is no substitute to going to Catholic mass! In fact, the Catholic church frowns upon a Catholic participating in less-than-fully-present Christ in the Eucharist, and not eating Christ's body as "bread of life" and as a sacramental means of grace. This is extremely relevant to you since Baptists is one of the groups farthest away from Transubstantiation (compared with Lutherans, for instance), because in the Baptist understanding, Communion is also NOT a sacrament, but a symbolic observance (as in Baptism). It may be a means of grace but NOT through the elements but through your spiritual communion with Jesus by remembering what Jesus has done for you. So in a Baptist communion the emphasis is on what the pastor was saying for the sake of meditation and what the worship team is singing to help you get in the mood.
Thirdly, it's about the Church as Sacrament. Being a Catholic means participating in a VISIBLE manifestation of the spiritual FACT of the body of Christ, i.e. the Catholic church as Christ's instrument (CCC 774-776). That's the logic behind why everything is public and visible in the Catholic church, including:
- How your sacraments (Baptism, Confirmation, Marriage) are recorded in a parish record somewhere. A Catholic priest I talked to takes pride in the Catholic church record keeping system where a parish can request the record from any other parish in the world. In this sense, your baptism IS already public record, as your marriage.
- How seeing the Eucharist is literally seeing Christ and eating Christ. Hence Eucharistic adoration.
- How hearing absolution from a priest literally hears absolution from God Himself because the priest acts in persona Christi.
Therefore, a Catholic priest I talked to when I asked whether I can be BOTH Protestant and Catholic, he said I can only do that IF I carry myself as a Catholic in the Protestant church in a VISIBLE way, by
the above logic of the Church as Sacrament. This means 3 things:
- Fulfilling my Catholic obligations
- When relevant, people should know that I am Catholic (i.e. don't be a crypto-Catholic / closet-Catholic)
- Do not participate in communion at a Protestant church since it means very differently
So if you are comfortable announcing among your Baptist friends that you are ALSO a fully practicing Catholic and for that reason you are refraining from Baptist communion, I suppose you can keep your identity of being Catholic.
By the same logic, the Catholic church also frowns at your believer baptism because it's already "public record" that you are baptized and you can only be baptized once. But if you intend to be BOTH Baptist and Catholic and thus need to be baptized at a Baptist church to participate further, you must consult with your Catholic priest to receive a possible dispensation.
Fourthly, going to Catholic mass is NOT a social act, but for praying together in the liturgy taking part in the Lord's Supper. The meaning of going to mass is COMPLETELY different than going to a Baptist church service. There are DIFFERENT ways to be more socially involved, to learn about God and Jesus, and to deepen your spiritual life in the Catholic church. Ways that you may not know or that you have not tried.
TO BE CONTINUED
Key difference between Baptists and Catholics: baptismal regeneration
Since your question is about baptism in Catholic vs. Baptist traditions, I highly recommend listening to Baptist scholar Dr. Gavin Ortlund debating Catholic apologist Trent Horn about baptismal regeneration in 4 videos:
Dr. Gavin Ortlund's initial presentation: Baptismal Regeneration: Responding to Common Arguments (Mar 14, 2022)
Trent Horn's response: REBUTTING Gavin Ortlund on baptismal regeneration (March 29, 2022)
Dr. Gavin Ortlund's response in 6 questions: Trent Horn and Gavin Ortlund on Baptism (April 27, 2022)
Trent Horn's fuller answers to the 6 questions: DIALOGUE: Answering Gavin Ortlund's Baptism Questions (May 9, 2022)