The brake lamps and the 3rd light are run off individual circuits. So since neither one of those circuits are activating and the fact that the remote start isn't shutting down with the brake pedal as well, then the problem is either the switch not being adjusted properly or there is no feed going to the brake pedal switch. You should have a 20a fuse that supplies voltage to the brake switch. Once you press the brake pedal and the switch contacts close, voltage flows to the stop lamp circuit which is protected by a 10a fuse and to the 3rd brake circuit, which is protected by its own 10a fuse also. So the only logical problem that can keep both circuits and remote start shutdown from being operational at the same time is voltage supply to the brake switch or mechanical operation of the brake switch itself. The stop lamps are grounded seperately from the 3rd light, so I wouldn't think both gnds went bad at the same time.
What I would do is unplug the brake switch and verify that you have a constant 12v supply on the supply line. If not, the most likely scenario is a bad fuse under the hood or a break in the supply line somewhere. If you do have 12v, then either the switch is bad or the plunger isnt adjusted right.
