The analogue of the Künneth formula for Borel $G$-equivariant cohomology can be obtained as the Eilenberg–Moore spectral sequence of a pullback
$\require{AMScd}$ \begin{CD} (X \times Y)_G @>>> Y_G\\ @VVV @VVV\\ X_G @>>> BG. \end{CD}
Alternately, one can construct a "geometric resolution" of one of the $G$-spaces and then obtain it as a Künneth spectral sequence associated to a natural filtration. Hodgkin's construction of the Künneth formula for $G$-equivariant K-theory proceeds from similar considerations. This is part of a more general strategy which given an equivariant cohomology theory $h_G^*$ and $G$-space $X$ constructs an "$h_G^*$-free resolution" $(X_{i+1})$ of $X_0 = X$ out of cofibers of maps $X_i \to Z_i$ such that $h_G^*(Z_i)$ is free abelian and $h^*_G(Z_i) \to h^*_G(X_i)$ is a surjection. One also needs to find "enough" $h_G^*$-free objects and this question and the freeness of such a resolution is dependent on the cohomology theory.
The Chern character $K^* X \to H^*(X;\mathbb Q)$ induces an equivariant Chern character $K^*_G(X) \to H^*_G(X;\mathbb Q)$, and I believe this should induce a natural transformation of Künneth spectral sequences if we can show that there are "enough" spaces that are both $K_G^*$- and $H^*_G(-;\mathbb Q)$-free.
I would like such a map and imagine this has been done somewhere already.
Does anyone know somewhere this is carried out?
