The point of contention is that people convicted of serious crimes can no longer legally change their name. These groups say that hurts transgender rights.

First, this has to affect almost nobody. The number of people convicted of serious crimes is already very low, the number of those who transition after committing such a crime has to be astronomically low.

Second, you committed a serious crime. The reason this law is being implemented is that a lot of people were trying to hide their past from a simple Google by legally changing their name. This is a common enough problem that the government is having to take action for it.

Nothing is stopping the small number of people impacted from using a different name in casual life, many people don't use their legal name outside of legal situations. So I don't really feel like this is a problem.

Right now in BC you don't require any proof at all to get your gender changed unless you're a minor under 12. You just fill out a form and pay a small fee and it's done.

Maybe the best option here is an exception that requires doctor approval via transitional care just to avoid felons doing it maliciously. It's a bit more of a burden, but it's not too difficult.