<<how do you deal with a stove in a place like the Alpine Graden>>
Why would you camp in a tent on the Alpine Garden? I mean, there are reasons there are no trees there!
For the actual traverse, for which I would start out hydrated to the point of overflowing, I would leave my stove at Crag Camp and carry 3-4 liters of warm liquids with me. For a camp or bivouac, for which I would prefer to be in a snowcave to a tent, I would have 2 cans of Sterno (large size) per person and a small aluminum pot and metal Sierra cup. It's slow but better contained than an open-flame stove, plus the nights are long and you're not likely to be distracted by what's on TV.
Again, my ingoing assumption (and m.o. in my younger days) has to do with picking the day and not going in conditions in which you're likely to be pinned down. Once you're on the traverse, if the weather deteriorates, bail out, don't wait it out.
BTW on my first (and failed) Presidential traverse, I frosted 9 of my fingers and my face. We had a 3.5# Optimus 111B stove and 4 # of fuel for the monster. Stop the madness!