Hi!
Last summer I spent three nights in Besse and could easily have spent more time. It was really lovely, a cute small town in the middle of a region of lakes and extinct volcanoes. Far exceeded my expectations!
6 nights could be a bit much, but there is plenty to see and do for 4-5 nights.
In the same region, I heard good things about Orcival, which is a tiny village. And the small towns of Murol and Le Mont Dore also have their charms. But there is no need to change bases within that region around the Puy de Sancy and Puy de Dôme.
If you are really going for 6 nights then perhaps you could split (depends on your travelling style!). In that case, I'd keep 3 nights for the Puy de Sancy área (so, in Besse for example), and spend 3 nights further south-west in Cantal. I know it less, but the area between Salers and Murat is really beautiful.
Food wise, be aware that this part of the world is close to vegetarian hell. But the beef is reportedly excellent.