Can I eliminate 1 switch from a 3-4-4-4-3 way configuration?
Go ahead and yank it
Since this is not one of the required switches for the stairwell lighting, and a 4-way, it's pretty easy to remove.  Simply nut the two black wires together and the two white wires together after removing the switch, then put a blank single gang cover plate over the box, turn the breaker on, and hang your mirror.