-
No, we've done it last year in July and it was already dusty dry.
It depends of what kind of winter is going on up ther and how the spring will be. There is definitely a chance some of the segments will be muddy, I am thinking Colombardo and probably the second part of Finestre, which aren't really exposed entirely might retain some more water.
Again, if you're taking the Surly you should be fine with some knobs on your tyres.
Fat tyres with for the fun!
I'd go with the Krampus, yes there will be sections where you will want a sleeker roadie ride, like the Izoard or Agnello, but that would make all the strada far less fun (especially in the uncertainly wet conditions of May - snow melting and all that)
Plus you'll have a bikepacking setup anyway, just accept that this is about having the most fun, not smashing any time on Strava.
That should also sort your carbon frame vs plane doubt.