The isomorphism for $K_G^1$ in the question is right. see
Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin, http://jtopol.oxfordjournals.org/content/4/4/737.full.pdf+htmlLoop groups and twisted K-theory I Section. J. Topol. 4 (2011), no. 4, 737–798.
Section 3.5.4 and Appendix A.5