If it is mid to late May you are more than likely able to walk up to GSB Pass. The roads could well be open except for the few hundred metres across the pass. I was there at the end of May and that is what I found, though I suspect it depends on the year. You are more than likely able to get to Saint Rhemy - a very pretty village. Aosta is a great place to start otherwise.
Which direction? SdC - Aosta is a beautiful area walked and climbed there often when I was younger...much younger. Thinking practically I would consider starting from Torino (not my favorite city) headed SW to catch the Via Tolosana in Arles. Rome? The Assisi Way begins in Vezelay flows through or closeby Aosta and would allow you to miss some those high passes - lodging on both the Via Francigena and the Assisi Way can be austere. I have walked from La Verne through Assisi to Rome fantastic views art and architecture but can be a challenge logistically food lodging etc.
Seems like a good enough place, though I'd mildly suggest starting the walk upon disembarkation at the airport or arrival from a train, even though on first go that might just be to wherever you'll sleep there.
And then (less mildly) walking from there all the Way to Rome, not just some section.
There's something from completing a pilgrimage through walking every step of the Way that is easily lost from unnecessary compromise (health problems excepted always).