It's true that the rent payment depends on the previous transaction's signature, but if that payment (even if mutated) has made it into even one block, the wallet software that generates the rent transaction is able to figure out which transaction on the blockchain is responsible for the change it intends to use.
But the mutated version necessarily has the same outputs, right? How does mutating then make your rent payment invalid?
edit: To clarify, once the mutated transaction is on the blockchain even once, the wallet should be able to figure out what the correct transaction is now. True, when it's not on the blockchain at all, the transaction can get rejected.