It depends where you're thinking of staying Bob and at what time of year, today for example I spent a few hours leisurely birding (with Bittern of this parish) around the Col de la Colombière in Haute-Savoie (about 90 minutes from Geneva or an hour from Annecy), mammal-wise I saw a Stoat early on then together we saw Chamois, Bouquetin(Ibex) and Marmottes. Birds included 30 plus Griffon Vultures, four different Bearded Vultures, two Short-toed Eagles, Kestrel, Cuckoo, Tree and Water Pipit, Dunnock, Black Redstart, Rufous-tailed Rock Thrush, Northern Wheatear, Whinchat, Ring Ousel, Garden Warbler, Blackcap, Alpine Chough (probably Red billed Chough briefly too), Citril Finch, Linnet and Yellowhammer. With a bit more walking we could easily have added Snowfinch, Alpine Accentor, Rock Bunting, Wallcreeper and Lesser Whitethroat which are all present at this time of year.
Further south in the French Alps you can see more Mediterranean species as well of course and a bigger selection of birds of prey and vultures, the Gorges du Verdon area for example gives you a chance of Bonelli's Eagle, Egyptian Vulture and Cinerous Vulture.
The 2019 published Where to Watch Birds in France from Pelagic Publishing has details of all the good spots so that might be worth getting.
"Thought I saw an eagle
But it might have been a vulture
I never could decide" Leonard Cohen