A little more reading doesn't dismiss the possibility this might actually be possible. GenderGP aren't aware of the option, and invite me to try.
There are two options:
"EU directive route", state or private, you pay up front, they reimburse you up to the NHS cost.
"S2 route", state only, they pay the state directly, either all or 75% (depending on the country of treatment). The copay may be reclaimable.
https://www.nhs.uk/using-the-nhs/healthcare-abroad/going-abroad-for-treatment/what-is-the-s2-route/