We walked Florence to Rome in July 2015 (I'm not sure if that is recently in your book or not). This route will be considerably more challenging than your previous Camino walking Sarria to SdC. Additionally, there will be very few other pilgrims (we saw a total of 5 others). As far as safety, I think there are three concerns.
Statistically, probably the biggest threat would probably be motor vehicles. Most of the road walking was on fairly quite roads and I did not feel it any riskier than the CF or CP.
Second, would be getting lost. The walk between Florence and La Verna is often on relatively unused forest paths. Without sign posts in places. Sandy's GPS tracks saved us more than once, especially when forestry had obscured the path since he wrote. After La Verna, Sandy's path merges with other more established and very well marked pilgrimage routes. From that point on we hardly consulted the GPS.
Third would be human predators. These woodland trails are pretty isolated and I doubt cell phones work in all areas given the steep terrain (and resulting "shadows" where signals are not available). I am a big guy, and my wife and I walked together. We never had a moment of worry about our safety in this regard. Others might feel differently based on their own situation.
One last word of advice: Do not go in the summer. I've walked the mesetta in July and thought it was hot but bearable. July in Italy is much hotter; too hot for any comfort or enjoyment. My wife and I are hoping to go back and do this path again in either spring or fall. The country and people were great, the mountain around La Verna (not the monastery itself, which was overrun by tourists) is one of the most magical places I have ever seen anywhere in the world.
Buen Camino,
Jo Jo