“Biting the bullet” means for metaphysicians to say goodbye to nearly all classical metaphysics.
In the hour of birth the term metaphysics denotes a specific collection of Aristotle’s lectures. In modern language these lectures belong to ontology, epistemology and philosophy of science.
Since then, the range of metaphysics was all-embracing. Its followers claimed to have found concepts and principle which are universally applicable and valid – in particular to explain the non-physical domain of our world.
Kant’s Critique of Pure Reason showed for the first time what “Biting the bullet” means for metaphysicians: To accept the limit of pure reason and to restrict knowledge to the empirical domain.
Kant did not eliminate the domain of ideas, of course not. But he separated rational knowledge from reasonable guiding principles for philosophical work in general.
A second prompt for metaphysicians to “bite the bullet” was the turn in epistemology due to Popper’s insight: The search for ultimate justification of general statements is futile. Instead, focus on conjecture and refutation of hypotheses.!
The insight that ultimate justification is an illusion has been illustrated by the Muenchausen Trilemma, named and elaborated by Hans Albert.
I agree with the comment of @gs and consider the Muenchhausen Trilemma a theorem in epistemology with the consequence: "Biting the bullet will be inevitable in any metaphysical theory", as understood in the sense of classical metaphysics.