Is it possible to function morally outside the human concepts of good and evil?
From my perspective the basis of your question is flawed... so there is no suitable answer.
Your statement is with the belief that the concept of good and evil comes from humans alone. If I was to take that view - then there is no foundation to work from. Each one can do what is right in their own eyes without a concept of evil. (It's happened previously - Judges 21). What does it matter what I think, what you think, etc. Each one for themselves to get as much as they can out of this life - who cares if you can get away with it... (but even that is pointless - fleeting, etc)
Good and evil can only exist where there is a constant. A single and unmovable point of reference of which to go from, otherwise it flips at the whim of the observer. "If" the universe is an accident, a big bang, and clumps of whatever randomly formed what we see today - indeed everything is meaningless as you have stated - and there is no such thing as good and evil.
On the other hand - if there is a single and unmovable reference we can look towards - which there is with God - then likewise the question is still invalid - because their is no human concept of good and evil - it is defined by God almighty.
So as such - good and evil can only exists if there is a constant / single unmovable point of reference.
So from my perspective - your real question is invalid because the foundation of the question doesn't reside on a human concept. Rather the real question is - is the God of the bible the one true living God, or not. Only from that point of reference can you find your way to talk about good and evil.