OK, since this issue is likely going to start to happen to a lot of Solsticii and Skys out there as they begin to age, I went rooting through my storeroom to find my spare Heat/AC controller and pulled it apart to give folks more info on the bulbs.
First, to get to the controller, you'll have to remove your boomerang (or whatever dash piece serves in the Sky). The controller is attached from the back with four 7mm hex-head screws.
Then, as mentioned before, you'll need to lever out the back of the controller, which is held in with 10 clip-tabs: 4 top and bottom and one on each side. See the photo and the yellow arrows indicating the 4 at the top...
Once that's off, the bulbs are shown here in the blue circles. 1/4 turn counter-clockwise and pull out - reverse to re-install. Once you've replaced the busted bulbs and want to put the back back on, make sure you align the fan speed knob arm with the switch arm (red circles) or you'll have no fan control
Finally, I still don't know what a correct replacement bulb is, but the following shows the dimensions of the bulb and its carrier, along with (measured) filament resistance and current. If someone finds an exact replacement, please add the details to this thread!