It's a very good question indeed. I want to answer the question and also give some specific examples. ;-)  It's a bit tricky with the wording "What is the Biblical Basis for defining atonement as at-one-ment with God."   Let's break it down a bit.  
It is Biblical to use this definition of atonement?  **Absolutely**. It is the most precise definition?? **No.** Let's break it down a bit. 
*"In Christianity, atonement refers to the needed reconciliation between sinful mankind and the holy God. This reconciliation is possible through the atoning sacrifice of Jesus Christ, as expressed in Romans 3:25, Romans 5:11, and Romans 5:19. Atonement is the Bible’s central message." Christianity.com
"In the King James Version of the Bible, the word atonement is only used once in the New Testament – in Romans 5:11. However, most other versions have translated the word “atonement” to “reconciliation” because that is the literal interpretation of the word.*"
We all know that there are many words with small, subtle shades of meaning. In most cases, we can use synonyms to give the same meaning, but in some contexts, a word with similar meaning just doesn't work. In a specific context, the word would work fine, but in another specific context it evokes a totally different sense and feel.  
The example is that when people come to faith in Christ, we are "Justified".  Christ is our advocate/ our defense attorney, and his father is the Judge and so God looks at us, and we are "declared righteous" by the work of the substitutionary payment of Christ.    Whew. This is a lot for a new believer to understand, let alone explain to someone else. The real, best meaning is literally "to be declared righteous", but people need a simple way to understand and remember -- [Child-like faith] so we use "Justified=Just-as-if-id-never-sinned.  In God's eyes, I have been declared righteous, so it's just as if I'd never sinned.  
This is the same as with the phrase "At-one-ment". 
In the Old Testament - the most common use is connected with the Jewish Day of Atonement. For Jews, this is still the most holy day of the Year.  The High priest would enter the Holy Place and sacrifice goats, which would be the payment for the sins of the people. This special holy day involved the scape goat, in which two goats were used. They cast lots and one goat was set free, and the other was the ransom which was sacrificed in place of the people. 
Other dictionaries say - making amends, to pay for a wrong or sin, usually by a sacrifice, etc, but the common theme is that there is a payment, and in scripture, we see the majority of time this payment was the blood of the sheep or goat. It is by this payment that we are reconciled to God. **It's not simply a formal declaration, but a life is given for another.  
So through Christ's death- his blood in place of mine, he was a the perfect Passover lamb, who was killed on Passover at the very time the perfect lambs were being slaughtered in the temple.  Through his payment we are Justified and reconciled to God. Once we were enemies of God, now, we are adopted, joint-heirs and made-at-one.  
If you can remember reconciliation, then great-  use that. 
When we look at the word, "Atonement" we can break it down At-one-ment and it's simply a much easier way for most people to understand and also remember the concept.  I think someone saying it's dangerous or in any way implies that it's something I did [works based] is totally off base and without merit.  It's reading something in that does not exist in the word or the definition.  

Hope that helps.