That's not how it works. In the event of a hard-fork, when the two forks adopt different rules about what blocks are valid, the actual users of BTC still decide which rules to follow. If a miner produces blocks that are invalid according to the rules that are adopted, the blocks that it produces are just disregarded. Thus, the hash rate competition between miners is only within each fork.

If you are talking about denial-of-service by producing empty blocks or blocks that exclude certain addresses, that's another matter...