Genny will probably be your best bet. If the cave has any wide shots, you'll never get exposure, even if your going for a dark look. If your camera is 320asa (average for video) you'll still need 1-2 thousand watts to get to a dark image on screen. After that you probably want hot pools of light and edgelights, which could push your draw past 5000w. Also you might want to look into renting (genny and lights) grip and rigging in a cave isn't easy, if no walls/ceilings/floors are plumb and level, so having professional light/grip might alleviate a lot of headaces. The least you can do is rent a bunch of rocky legs or lazy leg C-stands to make uneaven terrain easier. Also invest in a milk crate of wedges and cribbing. it won't cost much but it will save you many times over. Also get black stingers, very easy to hide in a dark cave. You can run power right through frame without anyone being the wiser.
Also think if you can modivate your cave light by flashlights the charecters carry, that might make lighting a bit easier.