OK, not a huge budget, and coming from Munich, heading east to Prague at some point. The answer is obvious- somewhere in either Bavaria, or Tyrol or Salzburgerland in Austria. You could head directly south to Garmisch-Partenkirchen, Mittenwald or Lermoos-Ehrwald. Each of these towns offers endless hiking opportunities and plenty of eating options. I particularly like GaP in the summer because they hold nightly free concerts in the beautiful Kurgarten. Further east, Berchtesgaden is popular, although the eating options aren't as extensive, and most of the best hikes start at trail heads a considerable distance outside of the town nucleus.
I like Obertauern in Salzburgerland in the summer because it's high altitude offers as an escape from the heat waves that occassionally afflict Germany and Austria. However, if you're looking for a tradition-looking Alpine village... this ain't it. The town was purpose-built as a modern ski resort. Most of the hotels and restaurants close for the summer, but the few that remain open are quite opulent and offer full board for pretty cheap off-season prices.
I've only visited during the winter ski season, but Kitzbühl looks like it would also be a pretty good location for summer hiking.
The only location of which I am aware that can more of less guarantee summer skiing is Zermatt in Switzerland. Very far out of your line of travel and very expensive. Hintertux in Austria has summer skiing some years, but it's unpredictable and based on winter and spring weather conditions.