Some googling has led me to an answer to the second question, at least: $H_G^{\bullet}(X, \mathbb{Q}) \cong H_T^{\bullet}(X, \mathbb{Q})^W$ holds for any $G$-space $X$ and the argument is essentially the same, and I learned from a paper of Harada, Landweber, and Sjamaar that $K_G^{\bullet}(X) \otimes \mathbb{Q} \cong (K_T^{\bullet}(X) \otimes \mathbb{Q})^W$ holds at least for $X$ compact (actually we only need to invert $|W|$). So this is encouraging, although I don't know how similar the proofs can be made.