When it comes to biometrics, why is it bad if I decide on the spot I'm not okay with it, as opposed to having decided as part of an institution?

It shouldn't, but the organized religions have largely held their ground on principle, when others have given up rights for broken promises of safety, or equality, or "for the greater good," but most of all, because they think it'll only apply to others or be used in a limited scope.

And that's never the case.

How is that distinction constitutional? Privately held beliefs are illegitimate unless they are held with a registered institution?

There often (not always) is no functional way to tell the difference between a belief made up on the spot and a “religious” belief.