SML,
Is your tail light assembly from a GM Pickup? I think mine are but, a slight variation of it. It's the exact same style, shape, and contour. Now, if yours is like mine, I have THREE double filament bulbs back there. Yep, even my back up bulb is a double filament. But, obviously it only uses one of them.
By the way, isn't using an "AMBER" tail light as a BRAKE LIGHT illegal? I thought I read somewhere it was, not sure. Talk about being lazy, if your installers did that, man, those were some inept "techs" if you want to call them that.
But, back to your issue. Yep, as one poster suggested, yes you could simply run another wire for activating that third brake light and, for the most part, it would most likely work just fine and the old one would not create any problems even if, is was still "Hot" up to the point where it's broken, providing it is broken and not just a tripped relay, diode, etc.
But, I'm with other posters and yourself. While it's time consuming, and, many times a serious P.I.T.A., tracing down an electrical fault to the point of origin is, for the most part, the correct way to attack this issue. Not fun, but, one thing, you'll get some nice gratitude when you find the problem and say, "Dam. it was only that? That's what I had to contend with when I put on my divers suit and dove into the broken wire situation in the J-1939 Data link problem that crippled all of my gauges, cruise control, and exhaust brake in one fell swoop. A single, broken, 16 GA wire, inside a connector, mounted to a bracket, up over the air dryer.
Talk about a pain. Anyway, keep at it partner. I may just pull one of my tail lights just to see the wiring from the third brake light and where it goes to, since our coaches are close to being a sister ship to each other. An '04 Itasca Horizon 36GD with the 330 CAT.
Scott