Can equivalences be strictified to isomorphisms?

I'm surprised that I didn't notice this immediately, and that no one else pointed it out either.

Every category is equivalent to a skeletal one. Therefore, every bicategory is (bi)equivalent to one whose hom-categories are skeletal. But in a bicategory of this sort, every 1-cell equivalence is an isomorphism. Therefore, every (braided, symmetric, ...) monoidal bicategory is monoidally equivalent to one in which all the 1-cell equivalence constraints are isomorphisms.

This is definitely an instance of the "whack-a-mole" aspect of coherence that Peter mentioned, though, since we can't expect to make a bicategory both strict (i.e. a 2-category) and locally skeletal. So it doesn't answer the "one way to make the question precise" that I asked, but it says something about the imprecise version.

It appears that the answer to the corresponding question for symmetric Gray-monoids is yes; this was shown by Schommer-Pries and cleanly reformulated by Gurski-Johnson-Osorno. Nick says that he thinks the braided case should work similarly.