I'm glad you defined what it is that is being asked to be proved - "real Christianity (the having a relationship with God by the mediatorial work and ministry of his Son, Jesus Christ)".
The difficulty of proving a relationship (in any sphere of life) is that claims may be backed up with evidence, but evidence, in itself, is not proof. I may claim to relate personally to my husband, signing myself as "Mrs. So-and-So", flashing my wedding ring to doubters, and waving my marriage certificate in front of deniers. Three strands of evidence that are in agreement certainly, but then if someone pointed out that I also have a certificate of death of my husband (which I failed to mention in my attempts to prove I was married), then my previous evidence would not count. I could only claim to have HAD a relationship with the man who once was my husband. That would also apply if I had a certificate of divorce from him. It may then look as if I had not been able to come to terms with no longer having a relationship with him, pretending to myself and others that I still related to him.
Now, if that's a complex situation with regard to proving a relationship with a husband, it is far more complex trying to prove one has a relationship with God! For a start, it assumes the existence of a God whom a sinful human can enter into relationship with. So, I assume your question does not ask about that. We can take the reality of God's existence as understood. And as you state that it is by the mediatorial work and ministry of God's Son, Jesus Christ, that we experience this relationship, then it becomes clear that belief in God's Son being the man who mediated for sinners on a cross is also taken as understood. Those things must, of necessity, be the foundation of any claimed relationship with God.
This means that anybody not believing in either or both of those foundational points can never believe another person's claim to truly relate to God. And it would be a monumental waste of time trying to convince that non-believer with logical proofs. It is only faith that counters a debate about Christianity by bringing the doubter into a living, experienced reality.
Those two foundational points for being in relationship with God require faith on the part of the person making that claim. It is impossible to relate to God without having faith in God's existence, and faith in what his Son did to open up the way to please God, personally. Nothing of that can be understood by a person not having true Christian faith. Even if they see a Christian living a transformed life and appearing to be all that a Christian should be, that will not constitute proof, for the doubter would say non-Christians can live transformed lives and be admirable people. And even if a Christian should present logical evidences for the existence of God and Christ, the denier will dismiss all of that because they have no idea what saving faith is.
All the Trinitarian Protestants I know would understand everything I've explained, and largely concur with it. Unfortunately, there are others who may agree with all the theology of Trinitarian Protestantism yet not have that personal relationship with God through Christ which we speak of, because it's just an intellectual assent and not putting faith in God and Christ. Maybe a bit like being happy to live in the same house as one you admire, but not being legally 'married', not having taken the step of commitment?
Anyway, here is a quote from a Trinitarian Protestant book on what saving faith is. It begins that section by saying:
"The grace of faith, whereby the elect are enabled to believe to the
saving of their souls, is the work of the Spirit of Christ in their
hearts... the principal acts of saving faith are, accepting,
receiving, and resting upon Christ alone for justification,
sanctification, and eternal life... Christ is both the author and
finisher of our faith."
"Faith denotes that aspect of change whereby the soul turns to Christ
and experiences supreme attachment to him." (The Westminster
Confession of Faith for Study Classes, G.I. Williamson, pp 96-97)
The book then goes on to use an illustration to explain why some who claim to be Christians, relating to God through Christ, are not yet in that relationship, never having actually turned from death in sin to life in Christ:
"We might call this 'spectator' religion. Christianity is, to them,
like a great stage drama - they know the lines by heart - and it moves
them deeply every time they see it - but they never become 'part of
the act.' This is not conversion. And it leads only to sorrow and
death." (Ibid. p 98)
I hope this explains why proving Christianity to be a relationship with God through faith of Christ cannot be done until a person tastes and sees that the Lord is good (Psalm 34:8). And, "Kiss the Son... Blessed are all they that put their trust in him" (Psalm 2:12). Or, "The proof of the pudding is in the eating", to put it in lay terms.
But it takes faith to reach out to the Son, to kiss him, spiritually speaking, to put their trust in him. Until a person does that, it's all just talk and argument, the words wafting away in the wind.
Therefore (as you say), "...the 'proof' of God's existence and the 'proof' of real Christianity (the having a relationship with God by the mediatorial work and ministry of his Son, Jesus Christ) is available to faith and to faith alone."