I'm working with globular operadic higher categories (with the Batanin/Leinster definitions) and ending up working a lot in the categories $P$-$\mathrm{Alg}$ of algebras for some globular operad $P$, together with (literal, strict, algebraic) morphisms between them. A familiar case of these is $\textrm{Bicat}_\textit{str}$: the category of bicategories and strict functors between them.

It would be very handy if there were some kind of mapping space constructions in these categories --- that is, some reasonable monoidal closed structures on them. Does anyone know what's been shown, either to exist or not to, either in the general case, or (more likely) for $\textrm{Bicat}_\textit{str}$?

(The most obvious specific candidate, of course, is Cartesian closure. However, even $\textrm{Bicat}_\textit{str}$ fails to be Cartesian closed: chasing through the Yoneda argument that shows what a Cartesian closure would have to look like if it did exist leads one to a counterexample showing that the product doesn't preserve pushouts. The next best hope would presumably be some sort of Gray tensor product; this is where I've not yet been able to find anything further on the closure question.)