I agree that the split 2 nights Puy Mary / 1 night Le Puy is probably better.
The hike up Puy Mary is surprisingly steep. There are stairs, but it is not an easy hike! Gorgeous view, though. If you happen to look for something more mellow, and less busy, Puy Violent is a great hike too, accessed from a car park up a narrow road just east of Salers. But that's if you do stay in Salers.
It is true that Salers adds about 1.15 hour of driving vs. Murat, going through the pass at the foot of Puy Mary. Total time to Le Puy is probably 3 hours from Salers. If that sounds like too much and if you are not finding anything you like in Murat, perhaps have a look in Cheylade? It's a small village, but the valley is gorgeous.
Also, not all apartment rentals use Airbnb, and besides apartments there are many B&B's in the area (you then need to eat out, but for 2 nights that's probably manageable?). I find Google Maps is a great tool to find such non-hotel accommodation, since it links to each property's details when available.