Oh, I just noticed, you have a multimeter. Yes, you can use that to make sure the switch is functioning. One terminal on each end is common. Meaning, that as you move through the postions, all the other terminals on that side, connect to that terminal. Clip one MM lead to the common, and run through the positions, testing each terminal. Then do the same for the other side of the switch (common terminal is diagonally opposite the first one). If your switch generally works, then it could still be your wiring of the switch. I just can't tell from here. You're trying to wire it as a simple, three way stock Tele, right? If so, you know your volume pot through the jack is okay, because Neck alone works. So, continue to isolate by testing bridge pickup for failure, then the switch.