.... or you could avoid the problem altogether and say 'He couldn't get down'.

Both are possible. It depends where or who your point of reference is. If you are looking at the situation from the point of view of the person in the tree, then it's 'go down'. If you are describing the situation from people on the ground, then it's 'come down'. This is always the case with come and go - it depends on the point of reference.

Perhaps these examples might make it clearer:- After climbing the tree, he couldn't go down because there was a dog at the bottom and he is scared of Dogs After climbing the tree, he couldn't go down and continue his journey After climbing the tree, he couldn't come down so we were delayed After climbing the tree he couldn't come down so that's why I called you In your examples, "Go" somehow implies leaving or "going" somewhere and, "Come" implies a return to a previous situation or to the way things were before. I hope that helps.