Health: Actually, a Medical/Dental school could admit someone without an undergraduate degree, and in fact many have programs leading to combined BS-MD degrees. The requirement of an undergraduatge background is “law” rather than “minhag” only for law and then not in all states. However its a very strong minhag.
Somewith a BTL combined with the correct coursework (available in many places that offer one year pre-med programs for people with degrees in other subject) would be adequate. Of course you would have to be especially brilliant with fanatstic grades in the science courses and great scores and a fanatstic resume – but it can happen. It’s easier to go to a regular college to take the premed courses as part of an undergraduate program. A BTL trying to to medical school would be no worse off than an English major or Jewish studies major.