It may also be worth contacting Peter Johnstone now, who may have a copy.

I would like to second the question by @DavidRoberts (hopefully generating another notification).

It may be irrelevant (I haven't the time to look more carefully now), but there is a more general way of composing monads than a distributive law: namely a wreath (§3 of The formal theory of monads II). It may be worth checking whether there's any relation to your situation.

Thanks, I agree with the conclusion now. I think it is essentially the statement of Theoreme 4.4 of Auderset's Adjonctions et monades au niveau des 2-catégories. I don't necessarily disagree that the condition I'm asking for is generally not the most appropriate condition to ask for, but it does seem like a natural one to consider (at least from a historical perspective).

Could you spell out why $\mathcal K$ admitting a terminal adjunction inducing $T$ for every monad $T$ is the same as admitting the specified right adjoint? Aren't the morphisms of the 2-category of adjunctions commutative squares, whereas the appropriate morphisms of "adjunctions inducing $T$" are commutative triangles? It's not obvious to me that the squares are forced to be trivial in this respect.

@IvanDiLiberti: thanks, I forgot about that paper – I've updated my answer to include it.

@SimonHenry: thanks for pointing that out. I found essentially this question in the context of geometric morphisms. An answer to one will most likely answer the other, but I think it's useful to phrase it more generally, so I'll keep my question open.