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.