There must be a threshold above which making an engine move is allowed. Because if there's a checkmate move, it will be the best engine move and needs to be allowed.
My immediate first reaction was also that it would be interesting to have a variant that is the same except you are allowed to checkmate, except then I realized the recursive nature of how board positions are evaluated makes that problematic. For instance, if there's a mate in 2, the first move of the mate in two is now certainly the "best move". Creeping up on a checkmate without ever making the "best move" until the very last one might actually be harder than the win condition based on strangling the opponent described in the current rules.
It just seems like you're changing the objective of the game entirely to the point where it's only slightly related to chess.
It is an interesting variation on chess given the current state of tournament play, yet it isn't really a solution to the cheating problem since it is effectively a new game with a new end-state. But you are probably right about there needing to be some sort of threshold. While there the rules of the variation says that any move with equal scores is considered equivalent, I would imagine the players would need a very intimate knowledge of how the engine scores moves in certain scenarios.
So essentially this converts most mates in two into mates in one, but some become ties by repetition.