2 fixed terminology

Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a ("bilinear") monoidal) functor $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak) comonoid in the 3-category of tensor categories, bimodule categories, intertwining functors, and natural transformations.

To compute what the induced tensor product does to two given module categories you will have to "compose" the naive tensor product with this bimodule category. This can be computed by an appropriate (homotopy) colimit of categories. It is basically a larger version of a coequalizer diagram. This is right at the category number where you will start to see interesting phenomena from the "homotopy" aspect of this colimit, which I think explains the funny behavior you're noticing with regards to bases.

Finally, you may get a braiding by having an appropriate isomorphism of bimodule categories,

$M \circ \tau \Rightarrow M$

which satisfies the obvious braiding axioms. Here $\tau$ is the usual "flip" bimodule.

1

Here is one set of data that will be sufficient. To get the monoidal structure you don't actually need a ("bilinear") functor $C \to C \boxtimes C$. It is sufficient to have a bimodule category M from C to $C \boxtimes C$. You will also need a counit $C \to Vect$, and these will need to give C the structure of a (weak) comonoid in the 3-category of tensor categories, bimodule categories, intertwining functors, and natural transformations.

To compute what the induced tensor product does to two given module categories you will have to "compose" the naive tensor product with this bimodule category. This can be computed by an appropriate (homotopy) colimit of categories. It is basically a larger version of a coequalizer diagram. This is right at the category number where you will start to see interesting phenomena from the "homotopy" aspect of this colimit, which I think explains the funny behavior you're noticing with regards to bases.

Finally, you may get a braiding by having an appropriate isomorphism of bimodule categories,

$M \circ \tau \Rightarrow M$

which satisfies the obvious braiding axioms. Here $\tau$ is the usual "flip" bimodule.