Just cam around this thread today. Thought it was still worth replying. My throttle had way too much radial and axial travel and felt dangerously lose.
I fixed it toady by putting a stainless steel washer approx 1.5 mm thick into the throttle grip after I removed it.
The washer remains at the outer right end and limits the axial travel. After mounting the throttle again, I moved the whole handset with break lever towards the headlamp and kept some tension on the throttle grip - as much as it still rewinds properly and does not block. As a result there is little to none axial travel left. At the same time, the radial travel of the grip which seems to be biggest at the switch housing also reduced to an extent acceptable for me.
Yet the diameter of the throttle grip fits poorly to the handlebar, e.g. compared to my '11 Speed Triple 1050, which only costs one third of the FLHRSE4.
Sorry for my language. I am German, not a native speaker.