We have a couple of hotel reviews
here. A rental might be another alternative for you, and you can find resources and information
here.
But first you need to decide which area would be a good base for you, based on what you're interested in doing and seeing. We have a page of info on mountain towns and villages
here.. Once you narrow in on a town, then try googling the village, perhaps getting to the Tourist Office website. I think then you can look at possibilities for hotels in your budget. I'm not sure a pool is necessary as I don't recall it ever being really hot and some days it was very chilly.
My family spent two weeks in the Berner Oberland in July/August 2005, doing a 100-mile inn-to-inn hike across the Alps. We stayed one or two nights in many of these towns and villages, as well as in some remote inns. It was all absolutely lovely. I would have enjoyed staying an entire week in Grindelwald, Lauterbrunnen, or Kandersteg, for sure. You can find our report
here to get more info on the towns, accommodations and scenery. There's also a link to a photo album.
Kathy