Hey guys, thanks for the help and sorry I couldn't get back sooner. Finals just ended and I finally have time to check the voltages.
Here they are according to the R/R diagnosis thread:
bike off: 12.4v
lights on, engine off: 11.76
Engine idling: 12.71
According to the thread my battery seems to be undercharged and I'm not getting enough voltage when the bike is running which would explain why the computer has been rebooting. I went to check the fuse and to my surprise it was melted but not tripped. Also there was a bit of scorching in the fusebox. This fuse was changed out once before when I didn't notice any issues. Pictures here:
I'm going to go ahead and change out the R/R to a mosfet type. Edit: I'm going to go with the FH012 and the Triumph adapter harness as per Decosse's recommendation.
Thanks again, I really appreciate the help!