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fc-multicategories seem to generalize what I'm talking about slightly, by allowing vertical $1$-cells between objects as well. I suppose that that is a natural generalization: for example, in the profunctor case we can take the vertical $1$-cells to be the ordinary functors.
Thanks for this. One other interesting thing I noticed is that if $\mathcal D$ is a representable multicategory, then $\mathcal V$ doesn't need to be the multicategory that we're enriched over, since then we can rewrite the hom object in $[\mathcal D,\mathcal V]$ as $\int_{d_1,\cdots,d_n} \mathcal D(\mathcal F_1(d_1),\cdots ,\mathcal F_n(d_n);\mathcal G(d_1 \otimes \cdots \otimes d_n))$.
