So, did you actually see the broken cable? Reason I ask is before you spend money on lubes and stuff you won't need make sure it wasn't the grip housing that broke. That's what happened on mine and several others.
The heated grips on the R3T are a known problem. If yours broke and you replace it you might as well buy a couple as it will break again.
I use a Teflon based lube have done for years but on long trips carry spare cables (best insurance against breaking one is to have a Spare) and I use one of them lubing blocks as in Bull's picture still using the same one I bought in 1976
Yup, the replacement they sent me went on a shelf. If you have heated grips on an R3T (and use them) the plastic will fatigue and break. Doesn't seem to be a matter of if, just of when. Luckily I was close to home when mine broke.