Rode out there a few years back. Didn’t do the Gavia or Mortirolo though.
We stayed in Hotel Stelvio in Santa Maria, Switzerland (hotel-stelvio.ch). Was OK for a couple of nights stay. Recent Tripadvisor reviews seem a bit mixed.
We rode down the Mustair valley then onto Prato and up the Stelvio from the North East (the classic way you see in most of the pics). Then down to the Umbrail Pass and over that back to Santa Maria. The Umbrail is unpaved for about 3km towards the top, but it’s well graded and easily do’able on a road bike.
From Santa Anna you can also ride up the Umbrail and down the Stelvio into Bormio or continue up to the top of the Stelvio.
There’s also a loop that goes Santa Anna > Umbrail > Stelvio > Bormio > Liviginio > along the lake > through the Munt la Schera tunnel (there’s a bike shuttle bus in the Summer) > Fuorn Pass (Ofenpass) > Santa Anna.