At least you were spared the new stock response the other Labour MPs are sending. Here's one highlight:
While I do understand concerns about how abusive men may exploit this process, I think there are many means for abusive men to access vulnerable women and children, and I don’t think that trans people should be denied rights because a minority of people will seek to exploit a new process.
You'll find a lot of suggestions for replies here:
www.mumsnet.com/Talk/womens_rights/3964304-Help-me-reply-to-my-MP-My-head-is-too-tired-for-this
www.mumsnet.com/Talk/womens_rights/3964332-MP-response-In-shock
For your MP I would suggest asking about the interplay between the GRA and the Equality Act. The EHRC MADE clear in 2018, that when it comes to the sex-based exemptions and female-only provisions, there is a difference between a male who identify as trans but remains legally male and a male who identify as trans who is legally female after receiving a GRC.
The former can be excluded without explanation or justification - where men are generally excluded from female-only provisions, such an individual is excluded too.
However, there are a number of legal opinions that posit that once a male is legally female through a GRC, excluding such an individual from female-only provisions is no longer lawful discrimination, but becomes illegal.
Furthermore, even if this legal opinion is incorrect (and it may well be, we simply don't know yet as this hasn't been tested in the court), although it is not impossible to exclude such an individual, in practice it can be very much so. And while this can be done with some effort, once you open up the GRC process to 100 times more men, employing the sex-based exemptions will be made much more difficult (dealing with a handful of cases may be doable, dealing with a 100 may be beyond the capacity of most smaller organisations). Without first clarifying this issue, without first strengthening the sex-based exemptions (for instance by making them opt-out by default where single-sex provision is the better option rather than something you have to actively opt-in a d by clarifying that funding decisions must not penalise organisations that choose to provide a single-sex service).
The second thing I would point out to her is that the case for reform has not been made by the government or the Labour Party. Misinformation has been at the heart of much campaigning about this. What we currently have is being held up as best practice in other countries. The Labour Government of the time, which created the Gender Recognition Act, strategically decided to make only a medical diagnosis mandatory, but not a medical transition to meet what they expected to be the European Court of Human Rights' changing stance away from requiring the latter (that was an entirely correct assumption. In a landmark decicion in 2017, the ECHR held that making a medical transition mandatory before granting a legal sex change violates the human rights of those who identify as trans, but crucially making a medical diagnosis mandatory does not. They deemed this ti be an entirely reasonable requirement).
And unlike any other application to the state, an applicant can receive support from a civil servant. The Gender Recognition Panels were set up in this way to avoid applicants having to appear in person and today it merely rubber stamps all applications with the correct paperwork. No one has to justify themselves in any way. If for any reason you are rejected, you can try again after a six months wait. The fee can be waived if you cannot pay it.
95% of all applications for a GRC succeed. That success rate speaks to a system that works entirely as intended, without any issues and is better than everything else we have (if you look at visa applications, disability or any other benefits, the huge failure rates there would lead you to a very different assessment).
So make the case for reform based not on emotional appeals but the fact. Then let's look at what needs to be done to improve the system we have and address whatever actual problems people who identify as trans have with the application.