The isomorphism for $K_G^1$ in the question is right.

Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin,  [Loop groups and twisted K-theory I][1]. *J. Topol*. 4 (2011), no. 4, 737–798. 

Section 3.5.4 and Appendix A.5 


  [1]: http://jtopol.oxfordjournals.org/content/4/4/737.full.pdf+html