At that size I wouldn’t bother messing about with mitres personally, particularly if you’re not saw-cutting them.
You’re unlikely to get them good enough not to look…. not good enough. (No offence intended)
The frame as you say, is fake so no need for much more than a butt-jointed frame (two vertical stiles running full height, then two horizontal rails, similar to how doors are made)
Screw and glue the corners. Drive the screws in slightly and fill and paint.
👆 Top Tip – Screws work better through the wood than laid on top like that 😉
Hang up with mirror plates maybe?
For the map itself, if it’s lying fairly flat to the wall then I don’t see why that would change when adding a frame.
You could consolidate the edges a little before putting the frame on perhaps by taking the map down and applying something like narrow, white Gorilla tape to the back side of the edge. Keep it strong then.
Then cover with the frame innit.