When putting up CM in a simple 4 sided room, what's the best way to deal
with the last piece - i.e. does one use a coped joint on each end?

I was considering putting up the first piece with one end coped, and fitted
over a small piece of scrap on the adjacent wall, the scrap being a proxy
for the last piece. Then when I get to the last piece, slip out the scrap
and slide the real piece in there. Am a little concerned about it being too
tight, though.

Any tips would be appreciated.

Thats how I do it. I also tack it in with brads not finish nails. In addition I have found that too short a piece (1') tends to be worse than a longer piece, as a longer piece tends to follow any variations in the wall and/or ceiling making it a truer representation of the last piece to go in.

Scot