I really like the formatting you have provided here, it is very helpful and transparent. I can't seem to get a simple expression to work with it, however. My take away is that it's not going to be possible to do a double integral in this way. For example, for the following function f:
g/((g^2 + w0^2) (DiracDelta[\[Theta]-1]+DiracDelta[\[Theta]+1])
The inversions should yield cosines, however the integrals provide the errors:
NInverseLaplaceTransform::wrongexpr: The argument invFT[g,0.2] is not Numeric at fixed g .
I think the approach should be as you said in your previous comment. I need to conduct the first integral symbolically and then I can do it numerically for the second variable. Thanks again for your help!