Updated 05/25/2011: Save yourself time, money, and frustration by checking out my latest solution.
I went back to Monoprice.com to update my review of the HDMI switch that I’m using and while glancing at the product page I saw this:
Note: With most models of Televisions both displays will need to be powered on for the splitter to function properly.
I don’t know if that note was present before I bought the switch but basically it means the problems with it dropping the signal are expected unless both TVs are left on. As a result, this means I’d need to get a different splitter to get around this problem.
I’ve spent enough money over the past couple of months on this project. Perhaps in the near future I’ll follow-up with a new purchase but at this point I’m resigned to just swapping the HDMI cables for the bedroom and living room TVs directly in the DirecTV HDMI output. It isn’t as convenient but it should work.
Update: I’ve changed the connections. I’ve seen two very brief drops but both times it picked up again without having to do anything.