Can I get HMRC to acknowledge my gender without a GRC?

I just came out to my manager at work yesterday, and she sent an email today, while I'm not in the office, to let the team know (I asked her to do this). She has also been letting HR, Payroll and some of the other managers know, and is trying to do everything she can to make this as smooth as possible.

She just forwarded me an email from the head of payroll who has said that he needs me to send an email to him requesting him to change my gender on the system (which I have now done), but he also so on the gov.uk website that he should ask for my new birth certificate (which I dont have as I don't have a GRC)

Doss he actually need to see an updated birth certificate for anything?

And to the main question, can I change my gender with HMRC without a GRC?

I have read online that they need a GRC to fully update my records, and without a GRC they will still record my gender on my HMRC records, but my birth sex will still be recorded until I get a GRC.

Is this true? and is the only way to get this done to send them a letter or is there some other way?

I read some reddit comments talking about changing gender marker on Goverment Gateway, but I couldn't see any option to do that. I also.saw some people say that you can email HMRC Section D, but I couldn't find an email address for them.