If you have signal lights then a lot of the wiring checks out ok ,its not your brake light bulbs nor your wiring going back there.
you say you have 12v but when you add load it goes away ? Thats very possible , amperage and voltage is inverseally proportional to each other. I suspect you have a loose connection somewhere in the lighting harness. There may be 12v there until you turn on the lights or apply the brakes and then it opens. Check the back of the fuse box for a loose or corroded wire .