You seem to have some experience in the Whites so this might be all known to you but...
I think I might want to be humping some snowshoes depending on when in "late spring" you are going. April-early May? I would take them. June? Nah.
Also, depending on the day of the week you choose, like Saturday or even a holiday, I would feel just fine about not taking a bivy or bag, especially if the huts are open, even on a self-serve basis, where there will be shelter. I don't know if the huts have potable water or not at that time, but they are open around Mid-May. Relying on others in place of taking your own gear is a no-no, however, and I would take a poncho-tarp at least and a space blanket or something.
This has less to do with your gear list, but I would also think about hitting Flume first and go clockwise, regardless of the snow depths, as I find Franconia Ridge and Garfield Ridge FAR more manageable in that direction than in reverse. Also, after the South Twin climb, it is a significantly easier go the rest of the way. Psychologically, hitting the highest peak early in the day is helpful to me.
Why don't you give us the gear list you are initially thinking of and let us tinker. Cheers