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Thanks for posting
Do you have a voltmeter/ohm meter or multimeter? It would help so we can check some voltages. Even a cheap test light will help.
Let's try a few things first, the brake lights are supposed to work regardless of the other lights on or off but if they go off when the lights are turned on is not normal.
First remove the bulbs at the rear, the bulbs for the head and brake lights have two contacts in the socket, make sure there is no corrosion or burned contacts, check the white/black ground wires for corrosion and be sure they are screwed on tight. Look for any wires that may have burned oe damaged insulation at the rear, let me know and we'll go from there.
yes, the center brake light could be involved, check the bulb socket they have a tendency to overheat, try a new bulb, try to disconnect the light completely and see if the rear lights work normally, check the green wire it should not be shorted to ground, the white/ black wire is the ground wire, make sure it has continuity to ground.