The space of variables $D_{ij}$ satisfying the constraints of a metric form a polytope (or really a cone, since any scalar multiple satisfies it as well). In general, integrating functions over such objects can be (computationally) hard. Here's an article that talks about this in more detail (for the specific case of integrating polynomials over a simplex).
Caveats:
- you're integrating over a cone, not a polytope: maybe that makes things easier (although I doubt it)
- you're integrating a very specific kind of function and maybe some specific tricks work for that case.
- if you're willing to get an approximate answer, I'd suspect that something might be possible