I just want to make sure I understand the situation as described by the OP.
There are only two planes in the sky near some airport. Just two planes. And they can't work out something on the radio to the satisfaction of both pilots?
I haven't heard anyone say something like: I offered to cross final and do a right 270 turn and rejoin final behind the plane calling on final.
Or: Plane on final offering to do a 360 turn to allow base traffic to go ahead.
And did both planes have recognition, landing, or other intense lights on over and above strobe and beacon?