If a move reaches the 50-move rule and gives checkmate at the same time, does the checkmate take priority over the draw?
Checkmate overrides the 50 move rule.
The game is drawn, upon a correct claim by a player having the move, if:
a. he writes his move, which cannot be changed, on his scoresheet and declares to the arbiter his intention to make this move which will result in the last 50 moves by each player having been made without the movement of any pawn and without any capture, or
b. the last 50 moves by each playerhave been completed without the movement of any pawn and without any capture.
The player making the move that mates could claim a, but he of course won't do it because he is about to win. The player losing does not have the move after the checkmate, so he cannot claim the draw.
This interpretation is also confirmed by Geurt Gijssen, in his column An Arbiter's Notebook from 2006:
Regarding your remark about the 50-move rule: only the player on move is entitled to claim a draw, but he is not forced to claim it. So if the 50th move produces checkmate – the checkmate stands.
This suggests that, unfortunately, there may exist no common understanding as to what the rule in question actually means.
The irony is that players with opposite interpretations of the rule can perhaps agree that the answer you are now reading were wrong, insofar as each opposite player believes the rule in question to be unambiguous, but unambiguous in opposite ways!