Do I need a hatafat dam brit if I am already halachically Jewish and circumcised in hospital?

My mother is halachically Jewish, had me circumcised at hospital (thank G-d!), but did not raise me Jewish. I’m now in the process of reclaiming my Jewish identity, through a reform synagogue (my grandparents go to reform).

My rabbi originally suggested a hatafat dam brit however later insisted I don’t need it because I’m already Jewish and circumcised, that the covenant as written in the Torah is satisfied.

However my understanding is that within Orthodox and Conservative spheres it would probably be required?

I didn’t want to push it with my rabbi, but I feel as if I missed out on all the rites and passages of growing up Jewish, so part of me wants to do this if it is recommended or required. I can’t explain the feeling that pushes me towards the rituals.

It feels awkward to demand it and also awkward to show my privates to the rabbi… since I’m already Jewish is there a way I arrange to have it done privately by a rabbi I will never meet again or mohel?