How about the Dolomites
? If you get some via feratta gear then there will be dozens of routes open to you. The mountains don't rise much higher than 3000m but the scenery is to die for.
If you're set on Tirol, I hiked across the region last summer, through the Karwendel
Alps. The Karwendel has some awesome jagged rocky peaks, although they don't rise beyond 2750m, there are many 2500m plus peaks that can be reached by scrambling. The Zillertal have higher peaks up to about 3500m which are supposedly quite non-technical although they are glaciated. (I didn't climb any peaks in this area).
Both regions are near Innsbruck, although the Karwendel doesn't have much in the way of road access, it will be at least a day's hike to reach many of the peaks