A is a person who does not deem themselves to have full knowledge of God.
I should point out initially that most theists believe that they
don't have "full knowledge" of God. God is supposed to be transcendent, to exceed human knowing, and so on. Theists typically believe that God has revealed something of himself to mankind, but that revelation doesn't exhaust what God supposedly is.
As far as A is concerned, strictly logically speaking, anything could be true about God.
I disagree with that sentence.
Atheists/agnostics (your 'A') typically use the word 'God' to refer to whatever the religious believers in their locality are using the word to suggest. So the A's define 'God' as the T's do, and then express their own skeptical doubts about whether such a thing can be known by humans and/or about whether it actually exists.
If the meaning of the word 'God' was a complete blank, if as you say "anything could be true" of God, then the word 'God' would essentially be synonymous with
'the unknown' or something undefined like that. (Anaximander's 'apeiron' perhaps, primaeval chaos.)
But the word 'God' almost always has more meaning and connotation than a mere 'something'. God is supposedly a giant invisible
person, with awareness, intentions and passions like our own. God presumably has plans and those plans involve us somehow. God is supposed to be super-powerful and to be responsible for the creation and continued existence of the rest of reality. We relate to God personally, not just abstractly and conceptually. God is supposed to be utterly
Holy in some emotionally evocative but hard-to-precisely-define way, completely unlike a simple question-mark. The precise content of the word 'God' varies according to religious tradition, but additional content is always there.
T is a theist, claiming to have proper knowledge of God.
T wants A to believe P.
A does not find P acceptable.
Given A's stance that strictly logically speaking, anything could be true about God, on the grounds of what may A refuse P
Well, if T wants A to believe in God (your proposition 'P'), then presumably T won't just be hoping that A believes something about something. (A already does that, many times over.) In other words, the word 'God' is always going to imply additional content, typically derived from whatever religious tradition T happens adhere to. Belief in God extends far beyond accepting the reality of the unknown. Virtually every A out there will already happily admit that there are many very real things that he or she doesn't know about.
A's response to T's apologetics will depend on the specific claims that T is making and on A's judgement of their credibility and liklihood of actually being true.
without thereby damaging their chances for doing what is right in God's eyes?
That clause seems to veer away from the logical thrust of rest of the argument.