I have never been able to charge any of my drills with a 12V inverter, the chargers refuse to take the power and will trip out internally.
The best Drills I have ever owned has been from Hilti, but I don't recommend buying one for use around the house or campsite unless you are drowning in cash. If you break it under warranty they will courier you a repaired or replaced one in a couple of days at most. The service is amazing, the quality is amazing, the price is crazy. My TE60 cuts through concrete like butter.
The best drill I have found for the money is from Makita, I have also owned and used and used up a few dewalts, millwaukees, bosch and others.
for going up and down though any drill would do, keep the jacks well lubricated and free of dirt, obstructions etc, and this should be a very easy job, you aren't actually jacking up the trailer, just putting a leg down to stabilize it. The hand crank is good exercise though.