You can get a *lot* of light out of LEDs with 22 watts (even factoring in efficiency loss, or the possibility of a small motor turning a lens to get the watery effect).
You know already know W = I * V, so yes .18amps @ 120VAC is correct. Watt rating is a good 'low common denominator' when figuring power requirements across different voltage supplies.
So 22W / 12V = 1.83 amps. Round it up for inverter efficiency loss, say 2 amps/hour. You can run that load off the internal battery NP, even with the engine off for awhile.
That said, you'll need much more illumination to pass DMV. As a reference, a motorcycle headlamp is about 1,400 lumens. Just to pass headlight/taillight DMV reqs, figure a min of 500 lumens (equivalent of 2x bicycle headlights and 1x tail light). For the rest of your MV, figure at least 3,000 lumens, more like 5K-10K lumens for good-better-best WOW.
For a high power LED, figure about 100 lumens @ 350ma. I am getting upwards of 1000 lumens @ 1.4A with some of the newest Cree Star LEDs(XP-G2), using constant current drivers (very efficient).
In summary, on the high-side, figure a ballpark 14amp hours @ 12VDC or 200watts ish. For 10 hours/night, I'd recommend ~200aH deep cycle battery (or 2x 100aH 6V in series) or a Honda EU-1000 gennie running in efficiency mode. Don't forget to factor in 10-15% efficiency loss in your calculations. You don't want to discharge a deep cycle battery much past 50%, and not to 10V (tho I plead guilty to this).
BTW, if you were around in 2011 and saw Aurora the willow tree, we were running about 3500 watts. That lights up the playa pretty well.