That assume a rule that wasn't state.

You're assume if someone picks 'checkmate' and the next player picks 'check' the games is over and the checkmate selector loses. I assumed that it means you treat it like 'check' 'check' and continue playing. But neither is actually specified in OPs post.

But let's assume it's your rules. Then winning is easy, just never pick checkmate. Literally never. As soon as your opponent picks it, they lose.

It's a terribly designed game as described.