we rented a chalet from http://alikats.eu/ last year, it was excellent, nice big dining room/social area and I think it slept 16 (but that is a French ski resort 16 which isn’t the same as 16 in comfort) and got it at a good price. But they don’t seem to have summer prices up this year or the same chalet. Still worth dropping them an email and asking what they have available.
It really is important to pretty much ignore the numbers they claim a chalet can sleep, probably knock a 1/3 off, depending on the make up of the group, if there is a few kids it is easier to stuff them in the cubbyholes. And usually you can haggle them down a fair bit on price, although you may have left it a bit late for that, still even with the boom in mountain biking, they are still not as switched on to summer and deals can be done.