LeeL
....the Swiss Alps and the South Tyrol Alps are pretty amazing for hiking. Lots of infrastructure. you won't find much wilderness and be alone in the alps
South Tyrol, Austrian Tyrol and French side would be a bit less expensive due to the exchange rate and the cost of things in general in Switzerland.
Flying into Tyrol is either via Frankfurt then one train to the mountains or to Innsbruck direct (a bit more expensive for the Transatlantic flight). After you're there just rely on public transit which is amazing
1st to 2nd week of September has way less crowds, absence of bugs and generally pretty amazing weather.
If you rent a camper and go town to town you won't go too far wrong. Or you can stay at hotels where they usually package your nights stay along with the tram/gondola ticket and breakfast.
You can hike down low or if you're not averse you can take trams to treeline and hike to alpine and then bike down.
I'm usually reluctant to post biking content as it brings out the resident bike haters but here are links to bike trips we did. In all these areas it is permitted to bike on hiking trails. Often we would see 1 bike for every 10 hikers and everyone pretty much got along. The trails are slow and technical so perfect for hiking but there are also lots of smoother trails perfect for hiking with kids - and we saw lots of kids and families out
lots of areas covered in the links - you'll have to figure out where you want to go. Personally i would stick to one approx area and work around that area just to minimize driving as there's much to see