I've looked at that hut trip and not doing it was one of my regrets from my time in CO. Let me know when the gE trip gets scheduled.
This would be an amazing thing. What is the best time of year for these sorts of trips?
I would try to plan it late summer, after the kids are back to school, and before any possibility of snow. Like first two weeks of September. June might also be option to avoid precip. If you're a risk taker maybe latter half of September when the quakies are turning. I think July August you could get thunderstruck off the mountain almost any afternoon.
Passes are just opening up, and I saw a decent amount of snow up high. A few people I talked to on the trail said they still had to posthole on a pass. This has been an unusually long spring with a lot of moisture, causing the snow to stay clean and cold, and, therefore, stay in tact longer before running off. Monsoons usually start early July/ late June, but this year has been off on that as well. We haven't had rain for about two weeks after getting quite a afternoon showers in June. It's suspected that it will pick back up later this week, but could be getting real dry here soon.
There is really no way you are gonna be able to plan around staying dry up there. The micro-climates can change in an instant, but your best bets would be just after snow melt (late June) or late August, through September. Ya gotta keep in mind that you will be heading into Moab, which is a desert, so avoiding the heat is probably a good idea. We usually don't start heading that way until late September due to not wanting to get cooked. Since you will be sleeping in a hut, you shouldn't have to worry about it getting cold at night. Even early October wouldn't be bad. If I had my druthers, I'd probably plan it in late September, early October. You'll miss the wildflowers, but have a much greater chance of good weather.