dg.differential geometry - Integration in equivariant K-theory

It sounds like what you are after is a theory of genuine equivariant K-theory (Which exists!)

If you think of a cohomology theory as a sequence of functors to abelian groups together with some properties and suspension isomorphism, then a genuine G-equivariant cohomology theory can be similarly thought of as a sequence of functors (indexed on the representation ring of G) together with "suspension isomorphisms" where you suspend by any G-representation, i.e. take the G-sphere with is the one point compactification of W (with its G-action) and smash with it. This is only the rough picture. The real theory is somewhat technical and develops the theory of genuine G-spectra. There were some comments about this here with some references. The punchline is that yes K-theory is an example of a genuine G-spectrum.

The next step is that you need to identify your K-theory with compact supports as the reduced K-theory of the suspension. I don't think this is too hard. Then your integration map reduces to the suspension isomorphism:

$tilde{K}^*_G( Sigma^W X) cong tilde{K}^{* -W}_G(X)$

As Michael and Kevin mention, under a suitable equivariant K-theory orientation hypothesis there will be similar integration maps for more general vector bundles, but that's another story.