Yeah, try registering as 'not allergic to penicillin' when that's not biologically accurate, and see how that goes....
But in fact the NHS [and HSE here in Ireland] will not actually say to a transman who wants a smear test 'Nah mate, you've made your choice, you've picked your team, now get lost', all the 'people with cervix' stuff shows how careful they are to ensure transmen get smear tests.
There has to be some degree of advance awareness of future complications like this when a woman decides to transition, and she has to take personal responsibility to give health care professionals accurate information, e.g. by NOT registering as male.