Unfortunately, this is the only area of DIY HRT I don’t have experience in, because I was already infertile before starting and had no desire to pass on my horrible genetics (a plethora of chronic illnesses). My understanding is that you would usually go about this at a dedicated fertility clinic, so if you were reaching out to a general practitioner’s clinic, then I’d try searching for a fertility clinic instead. Depending on how healthcare works in whatever country you’re in, you may need a referral from another doctor to the fertility clinic. I’m not aware of any other method of long-term gamete storage than cryopreservation.

Here in the US, there are even private options that allow at-home collection and pre-paid shipping to a storage facility, though I imagine that’s more expensive than simply going to a local fertility clinic (I don’t have any personal experience, though).

You will not be fertile on HRT* and possibly irreversibly infertile after a while on it. Storing sperm is the way to go, but I don’t know much about the subject. Where I am I can have my sperm stored by talking to my GP, but it sounds like you are somewhere where this has to be handled privately?

*don’t assume you are fully infertile. this can lead to accidental pregnancies. you will absolutely be less fertile and likely not fertile at all, but I won’t guarantee it