If you have power at the plug, but the light
isn't flashing when only the key is ON, I suspect a bad connection anywhere in the circuit from the TV's junction box to the light
that will respond when it sees the slighter higher voltage available when the engine is running vs when it's only on the TV's battery
I have found several times that a 7 pin TV plug that would make a light tester flash, would not power the trailers lights
until the 7 pin contacts were thoroughly cleaned.
I'd start by cleaning the contacts in the 7 pin connector and the checking the connections inside both sides of the connectors, then checking the connections to the TV's loom, then going back to the line in the BF.
As you have middle aged BF you may find a high resistance corroded connection almost anywhere.