If you think you may have battery cable/terminal problems that are preventing cranking, then fix that right away. We don't want to be confusing this thread with symptoms that are not related to the brake switch problem. How about Jack's comment, does this car normally require the brake pedal to be pushed to enable cranking?
Jumper 1 and 3 again (always, key-on...) and read voltage on those wires; you will have to pierce the wire insulation with sewing needles/pins to be sure you are reading voltage on the actual wire conductors. With them jumpered the voltage should be the same, of course. I don't know whether it will be the 12.7 or the 1.14, but am guessing both will go to 12.7 If both wires do have the same voltage, then corrosion is not likely to be a problem, as the wires have been proven to be adequately connected together just the same as supposed to happen through the switch.. By the way, the baking-soda will work in just a few minutes, no need for an overnight soak. I know things are tight where you are working, but try putting the baking-soda solution in a small jar and dangling the harness end into it.
Again, if the jumper (or the switch) can connect the wire conductors so that they have the same voltage, that is the best you can have. No use fussing about corrosion any more. I strongly encourage you to try the switch itself for this "jumpering", since the terminals in the harness may be bent/deformed such that a jumper makes contact with them, but the switch does not...
If we can be confident that the two wires are being connected together, and the lights still don't come on, then we will have to move back to the PCM connector that the wires end at and do similar voltage tests.