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Could you spell out a little more why this property is the same as essential surjectivity? (In particular, I do not want to assume equalisers.) I'm happy to assume $F$ is bijective-on-objects or even identity-on-objects if that makes things simpler.
Bicategories internal to 2-categories have been studied in Internal bicategories by Douglas–Henriques. Internal monoidal categories ought to be one-object internal bicategories.
@RoaldKoudenburg: thank you! Day and Lack cite a different paper of Lidner's, "Enriched categories and enriched modules", which had confused me as the result does not seem to appear there. It seems they cited the wrong paper. Thanks for digging this up!
Theorems 13.8 and 13.8* are assuming limit preservation, rather than proving it. In fact, he says that he's proving two functors are adjoint. I don't see that he explicitly proves that right adjoints preserve limits.
