one option could be to stay in or near Berchtesgaden Germany, just south of Salzburg, it really is so very close to Austria. the surrounding villages/scenery is like gimmelwald. ( just not necessarily up on the 'side of a mountain' as such) there are plenty of things to do and see in the area.
we recently stayed in Ramsau bei berchtesgaden ( germany) and Ramsau am Dachstein ( austria) both have very similar alp type villages and the surrounding towns have a similar feel. especially if you have a car. investigate the surrounding areas.
the big advantage they are both very much cheaper than switzerland. and some of the towns are comparatively un touched by tourists.
have you considered staying in a B&B. they are VERY reasonable in these areas. and you get to interact with the locals
if your family is into adventure then staying at somewhere like Grobming in austria, where you can go on the zipline and visit the nearby der dachstein cable car/ice cave etc. just a little further away from salzburg, but good highways between.
hope this helps.