One answer to Ben's question is to calculate the Hochschild homology 
of the category of $D$-modules $D_\Lambda(X)$ with fixed microsupport $\Lambda\subset T^*X$. The functor $D_\Lambda(X)\to D(X)$ has a continuous right adjoint, so defines a map $HH_*(D_\Lambda(X))\to HH_*(D(X))$. Thus given a $D$-module with $\Lambda$ microsupport we have its characteristic class with supports and its image in de Rham cohomology which is its usual characteristic class (but may be zero as you point out). Formally one can apply this idea in the other settings Ben considers, but one needs to calculate the
$HH_*$ of the categories of sheaves with supports geometrically, perhaps along the lines Tom indicates.

I haven't thought through the calculation of the relevant Hochschild homology - one needs to think through the properties of the adjoint functor above (enforcing $\Lambda$-support) which are worked out (in the coherent setting) in Arinkin-Gaitsgory, but I don't think it's easy.

But in applications one often sees microsupport enforced by equivariance - i.e., you might be considering $D(X/H)$ for some group $H$ with finitely many orbits (e.g., the flag variety with $B$ or a a symmetic subgroup, or general spherical varieties), so that $\Lambda$ is the union of the conormals to the orbits. In this case we are asking to calculate $HH_*(D(X/H))$ and the induced map (from pullback under $X\to X/H$) to $HH_*(D(X))$. Note that the category $D(X/H)$ recovers its nonequivariant cousin, the category $D_\Lambda(X)$, from its structure as a category over $BH$ so I think we can ``deequvariantize" this calculation to answer the question for these special $\Lambda$'s.

This is very geometric: by the results Ben quotes, the characters of equivariant $D$-modules are given by Borel-Moore homology of the inertia stack of $X/H$, i.e., of the stabilizers of the $H$-action. These are I believe related by linear Koszul duality to the characteristic classes with support that Ben is talking about (see the papers of Mirkovic-Riche).