I got a response from Kuka. The core issue is that the program I was running had non-continuous motion, which stops the motion interpolater and means RSI isn't able to run. If you can ensure you've got continuous moves within RSI, you're good to go.
We ended up sending the 'stat' byte back over to the PLC so we could pause the correction values while RSI wasn't running. When RSI does resume, it expects the input to be exactly where it was before it paused. Not zero. This took some guess and check work :-). We also noticed that if you specify a move with a tool change, that'll cause the joint position to go to zero for one cycle before it goes back, which causes an over torque. At this point that's a limitation I can live with and I'm not looking into it further.