The RBT 6 speed transaxle does, indeed, use a cable shifter and no shift gate. I believe it does not use the reverse lights switch that is used with the 5 speed's normal shift linkage. The back plate stud issue is a possibility though.
I also might have suspected a clutch release deficiency but as John can get it partially engaged (but pops out of reverse), it's probably a cable adjustment issue. I would speak with Corey Price, who recently acquired the cable shifter business from SACC, or Lloyd Butfoy, who designed the first version of the cable shifter, if I'm not mistaken.