On a '78 Series bass, I believe it is the case that if you plug stereo headphones into the 1/4 jack, you should get one pickup in one side and the other pickup in the other side unless the jack has been rewired for mono. You might want to take a picture of the wiring connections to the jack and post it here.
Try this, gently unplug the two pickup leads and switch them. Did the problem move to the neck pickup or is it still with the bridge pickup? If it's still with the bridge pickup then the problem may be with the pickup. If it moved, then the problem is not the pickup. Let us know what you find.