That does not sound like a complete short of the bridge pickup signal - in that case the middle position of the switch would be dead, regardless where the bridge volume is. Time for a structured troubleshooting process:
1. Put a guitar cord into the guitar jack. Clip one probe of the multimeter to the free jack's tip, and the other probe to the free jack's ring. Set the meter to DC resistance in the up to 20k range.
2. Turn all the controls up and
a) put the pickup selector to the neck position - what do you get? You should get 5k - 9k. Slightly higher is still OK but not significantly higher.
b) Put the pickup selector to the bridge position - what do you get? You should get 7k to 12k.
c) Put the pickup selector to the middle position - what do you get? You should get 3k to 6k, because both pickups are in parallel now.
Please do these measurements and post the results. That will be helpful to pinpoint the culprit.
Cheers Stephan