Let me save you some $$$$.
Step one, download this troubleshooting tree
http://www.electrosport.com/electros...t_finding.html and complete every step.
I am having a similar if not identical problem and I thought I needed a new R/R so I bought a model RR155 (non OEM) replacement from
http://www.electrexworld.co.uk/#1X0. It works fine and is half the cost of an OEM (Triumph) part but it didn't fix the problem.
It looks like the low voltage output is because of a bad Stator (a $400 part from Triumph with no non-OEM replacement available). I have temporarily "fixed" the problem by reducing the demand on the alternator to a level that the weak stator can sustain. How? I wired a switch on the headlight and I turn it off when I ride in bright daylight, I only ride in bright daylight. I like the look of the switch on the headlight bullet. It's identical to both the Trophy and the Tiger I owned in the '70's.
At my next oil change I intend to take the stator off and take it to an alternator / starter repair shop to see if they can rewind it economically.
Good luck and keep me posted on your progress. I'll let you know how I fare with a rewind if and when I have that done.