Alrighty, I have an update. Here is a bit of info that I got today. Apparently, the gent that I bought the bike from had it in the shop about a month or two ago to get the ignition switch replaced. He didn't know anything about the whole headlight relay issue either before or after the work, but they did replace the ignition as well as installing a new wiring harness. I assume, and we all know how dangerous that is, that the relay was replaced at the time.
That being said, I asked him if he was having to keep the bike on the battery tender just after that replacement or before and he told me that the bike had always been like that and he just thought that was how this particular bike was. In short, I think there was a problem, but he just didn't realize that there was a problem since it was being compensated by the battery tender.
What I am curious about is if there is any chance Triumph might cover this even though it is out of warranty since it had the issue during the warranty period.
Most likely not, but it doesn't hurt to ask.
Now to the bright side. I called the shop and while I was talking to the guy about all this new info I had uncovered, he told me that the mechanics just brought the bike back in from the test ride and it is good to go! He told me he would have to get into the notes to see what has been the problem, so I don't know what that is nor how much this fix is going to cost me yet, but I will find out in about an hour. Here is hoping for an inexpensive fix...:sad:
I'll post back on here later with what actually went down. Thanks all for the thoughts and suggestions!