let's get real simple here...
what does this rear fog light do?
every car I've owned, including my '49 Buick, had back-up lights. Nowadays they're usually incorporated into the taillight assembly, though in '49 they were separate.
so if the object is to provide more illumination behind you when backing up,,,,,
the wire to your existing backup light will have capacity to drive one of those 45 watt LED aircraft landing lights and it will be energized when you're in reverse.
Back in the days when cars had four headlights (1960's) I used to put 4522's in the inboard two for mega-high beams. But the factory wiring wasn't heavy enough for those 250 watt lamps so i added separate wire and a relay.
in lamp terminology PAR means the glass is round parabolic reflector, the two digits following are its diameter in eighths of an inch.
The GE 4522 landing light is a PAR45, ie 5&5/8 inch diameter.
An old quad car headlight is a PAR46 , 1/8 inch larger. The PAR45 with a piece of inner-tube wrapped around it fits right into an automobile quad-headlamp ring.
The 4522 lamp illuminated cowpastures a half mile away . If that LED landing light has anywhere near comparable lumens, it will really impress your friends.
one of those LED landing lights should give you no electrical problems, just you got to mount it. You shouldn't even need a relay.
but be sure it can't shine at other drivers. i LOATHE those Xenon headlamps in the hands of imbeciles who won't dim them.
over and out,,, old jim