What happens when the verification of a manipulated transaction fail?
Nothing prevents a miner from creating a fork which contains invalid blocks. However just because they are in a blockchain does not mean that everyone will accept them. All of the other full nodes on the network (including other miners), will receive the block containing the invalid transaction and reject the entire block as invalid. So that block will not become part of their main chain and neither will any block built on top of it, regardless of how much work it has. So a malicious miner could do that, but the invalid block wouldn't become part of the Bitcoin blockchain and every other full node would just reject it and ignore it.

The miner could continue to mine this invalid blockchain. But if no one uses it, then his coins on that chain are valueless and thus the miner would be losing money to continue to mine it.