Based on what you're saying it seems that the jeep side of the plug for the class C is not feeding the same wires as the Jeep itself if that makes sense. How is the Class C wired to the Jeep? Is there a box on the Jeep that the class C feeds into?
Once at the Jeep are there separate wires from the class C all the way back to the taillights? Or are the Class C wires terminated under the dash by the brake pedal?
Another thought. Maybe your Class C is miswired and is actually feeding the Jeep taillights and not the Jeep brake lights. Take a Jeep taillight lens off and check that the same filament lights when you have brakes on the class C and Jeep and bike rack alone.
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2001 Suburban 4x4. 6.0L, 4.10 3/4 ton **** 2005 Jayco Jay Flight 27BH **** 1986 Coleman Columbia Popup