It sounds like you're thinking carefully through your healthcare choices and looking for a practitioner you can trust. Biological or holistic dentists are fairly new on the scene and there is no agreed upon definition, standard or certification that I know of that indicates the additional training a dentist has taken to qualify them as 'biological.' To make the selection, I'd advise you ask around about this quandry so you can better assess the one you go to.
I know nothing about removing a filling from an abscessed tooth to let it breath nor of the wisdom of filling it temporarily. I imagine the dentists have a range of opinions about this so it doesn't make your decision easier. We do not provide any kind of medical advice here, sorry.
You need to be sure you don' t have an infection and, that if you decide to wait on the filling, that it does not put your tooth or immune system at risk. Tonsillitis and shaking? It sounds like you're in a weakened state so your decisions need to be taken with extra care. Do you mean YOU are shaking or your tooth is? If it's you, this sounds like you need medical attention immediately.
To find a dentist you can trust you might phone around to well known naturopathic clinics and ask at health food stores as well. Then you can at least start with them and ask them for as much clarification about their approach and why they call themselves biological as possible.
I don't envy you the search and I have waded into the world of dentistry when I or loved ones have needed care. It is a complex and evolving field to understand.
All the best,
PS Keep in mind the massive effect that your second dentist's opinion can have on you. If you are not in a medical emergency, you might look up EFT (Emotional Freedom Technique) to help you with the pain while you do what is necessary to decide what your next step is.