You've hiked it before? So you have an idea of the basic layout? Have you checked the phases of the moon? Last night was Waxing Gibbous, 97% of Full and I didn't need a light outside. Of course moon won't help you much in a whiteout. Personally, I'd just keep going up or down. On the wall at the visitors center at the summit, is a story of someone, long ago, who lost their way in a whiteout, they found the body very close to shelter. And yes, you won't want to fall in Tuckermans. Will you consider the trekking poles for this? I know you've done a big section of the Devils Path in the dark. Good Training.
There's a stretch of big rocks to clambour over near the summit. You'll want to be completely buttoned down for this section. Anything you drop will be lost. I'd suggest extra gloves and hat. Everything leashed like whatever you're using for route finding. Two even three lights like you were going into a cave. A coat and bag with lots of little pockets so whatever you need will be by itself.