I contacted Geoffrey Cruttwell with regards to this question. Here is his reply:

There hasn’t been any published paper on a tangent category of combinatorial species. However, the ideas can be found in a talk by my co-author, Robin Cockett here: Can you Differentiate a Polynomial?. The idea there is to show that polynomial functors form a Cartesian differential category (and thus a tangent category, since any Cartesian differential category is a tangent category). A particular example of this differentiation is then differentiation of combinatorial species (eg., as in wikipedia). We haven’t yet gone any further with these ideas, but there may be interesting developments if one attempts to apply some of the tangent category theory to this example (though it may be a bit tricky, as this example is perhaps more naturally a differential/tangent bicategory).

So, there is no differential category of species. Instead, there is a (putative) differential bicategory in which 1-morphisms (between particular objects) are species and the operation of differentiation on them is the usual differentiation of species. To see this, species should be represented as analytic functors. Then the construction of this bicategory should be similar to the bicategory of polynomial functors (which are a special case of analytic functors) which is described in the slides linked above.