The Raid Dolomite route would be my suggestion, from memory;
Day 1: Thonon-les-Bains -> Sion
Day 2: Sion -> Domodossala
Day 3: Domodossala -> Bellinzona
Day 4: Bellinzona -> Chiavenna
Day 5: Chiavenna -> Bormio
Day 6: Bormio -> Merano
Day 7: Merano -> Val Gardena
Day 8: Val Gardena -> Cortina d'Ampezzo
Day 9: Cortina d'Ampezzo -> Vigo di Cadore
Day 10: Vigo di Cadore -> Udine
Day 11: Udine -> Trieste
That's approx 100 kms a day, but on average the climbing ranged between 3000-4500 metres a day.
It was amazing - did it twice, first in 1998 and again in 2003. Would love to do it a third time but that'll have to wait until the kids are much older.
Livigno is full of hotels, don't know how many are open in the summer though.
The Raid Dolomite route would be my suggestion, from memory;
Day 1: Thonon-les-Bains -> Sion
Day 2: Sion -> Domodossala
Day 3: Domodossala -> Bellinzona
Day 4: Bellinzona -> Chiavenna
Day 5: Chiavenna -> Bormio
Day 6: Bormio -> Merano
Day 7: Merano -> Val Gardena
Day 8: Val Gardena -> Cortina d'Ampezzo
Day 9: Cortina d'Ampezzo -> Vigo di Cadore
Day 10: Vigo di Cadore -> Udine
Day 11: Udine -> Trieste
That's approx 100 kms a day, but on average the climbing ranged between 3000-4500 metres a day.