The first time I opened the TV up, before posting the message for this thread, the PIP board seemed slightly crooked. That easily could have happened while I was pulling off the case, or I may have just imagined it. I inspected the board for cold solder joints, used a meter to check some connections, and didn't find anything wrong. On the optimistic hope that it was just mounted badly, I re-seated the board, put it back together, and the set worked again... for about 6 hours. Then it went back to clicking.
I posted on here, and on Jason's and Ron's responses, I pulled it open again and re-soldered the LM317 and the surface mount resistors near it. That seemed to do the trick, it was working great. I figured I'd wait a couple of days before posting my results for the benefits of others.
Today, I've got a new problem, audio with no video. The menu works fine, so I'll have to start searching again to find out what's going on. I'll post my results on here again, along with the links that helped the most, in case it helps someone else out.