Sir goodalejason . . . . .
( Whut's a Throwingchicken . . .Version V3H ? )
Any how . . .confirm if that is being one single heavy BLACK wire coming from the lamp socket . . .which I would consider as normally being properly wire coded by use of a RED wire.
Now, of those two GREEN ended connectors coming from the battery, I will refrain from any referring to the SLOPED left one. The right one appears to be using spring pressure contacts OR an Euro contactor with its pressure set screws, being over on the non visible side.
Looks like you now need to get a DC voltmeter and determine the 12 Volts presence from the battery and MOST importantly, the polarities of the two end contacts. Mark the + polarity with a dot of RED fingernail polish / dope at an adjunct point on the GREEN plastic housing.
Whilst doing the confiscating of Mamma Cass's nail polish, you should have also plugged in her blow drier and turned on, and set to high heat and left the unit on and THEN unplugged the unit and carried it with you.
Meanwhile . . .back at the Throwing chicken, you need a ground wire added to that lamp which can be a split or ring terminal to connect to another wire(Or the wire end itself after having formed into a ring terminal ) that then can go under one of the two lamp socket mounting screws. That will then be your newly made negative LED lamp wire connection that goes into the battery NEGATIVE terminal.
With one complete lamp to battery connection now complete, you are now ready to make a quick SMOKE test by your taking of the blow driers naked AC plug and touching one of its blades so as to make contact to the battery connector RED (+) connection while the original BLACK wire from The LED lamp makes a VERY quick momentary touch to the other blade of the blow drier plug.
IF you then get a temporary white light for that duration, you are then ready to chance connecting the BLACK lead straight into the + connector for a few seconds to see if all seems well. Then you can assume that all is well and paint a RED ring around that original BLACK wires end to mark its + polarity.
Thassssssssit.
BUT I still . . .gots to knoes . . .whut is a Throwingchicken . . . . . cuz now, you is going to be havings to throw the WHOLE contraption !
73's de Edd