This amounts to saying doctors should be forced to perform procedures they believe are not in the best interests of their patients, go against conscience, or expose them to significant litigation risk.

All surgical sterilization procedures come with physical risk, and many other birth control does as well.

You think doctors should be forced to perform procedures they don’t want to perform?

The current version of the Hippocratic Oath says this:

""" I will apply, for the benefit of the sick, all measures that are required, avoiding those twin traps of overtreatment and therapeutic nihilism. """

Which does not include "except those I don't feel like doing".

The Declaration of Geneva states the following:

""" THE HEALTH AND WELL-BEING OF MY PATIENT will be my first consideration; I WILL RESPECT the autonomy and dignity of my patient; """

Which I read to similarly state that a doctor's own preferences as to the procedures is necessarily secondary.

At the very least the doctor should be required to refer the patient to a different doctor who is capable of performing it. This question is kind of similar to "Should pharmacists be able to deny birth control (or other medicine like vaccines) to people because of the pharmacist's personal religious belief?"

Except it’s not a question of the doctor’s personal religious belief or preference, but rather a question of professional judgment.

It is technically possible for a physician to prescribe methamphetamine or fentanyl. Many adults in the United States want to consume methamphetamine or fentanyl. (Sometimes both at the same time!) But it is very hard to find a physician who will write those prescriptions purely on the basis of “the patient asked for it”. Like it or not, some degree of paternalism is an inherent part of the practice of medicine.