The "does this answer your question" comment was automatically generated by MathOverflow; I didn't spot that it had been posted, sorry. You can ignore the part about profunctors, but the size issues it asks about are exactly the same as the size issues you ask about (an adjunction between bicategories is called a "pseudoadjunction" and generates a "pseudomonad").

@user984603: if you read even the introduction of the paper I linked, you would see it is answering exactly your question, and introduces the concept of relative pseudoadjunction. Cisinksi's comment is a corollary of the results of that paper.

Does this answer your question? Profunctors as a Kleisli bicategory

It's a relative pseudoadjunction.

@MikeShulman: thanks for pointing that out, I forgot to deloop $\mathcal V$. I've modified the question to involve enrichment in a bicategory $\mathcal W$ to avoid that confusion.

@TimCampion: thanks, I had just remembered that, came back to edit my question, and saw you had made the same observation :)

@DavidRoberts: I've clarified my question and corrected a typo: the relevant fact is that fully faithful functors are closed under pushout, not pullback.

@MikeShulman: indeed. The question then becomes, I suppose, "Can properties of such diagrams be deduced from existing theory (e.g. the theory of adjunctions in a 2-category), without having to reprove various results about adjunctions at this greater level of generality?".