Mountain itinerary ideas from Lake Constance - Switzerland or German/Austrian Alps?

Ok, first things first: Reichenau is between Radolfzell and Constance, so no need to do that on the leave.
Second: Depending on your experience with Switzerland and how old that is, it might definitely have become significantly more pricey. Priceyness is by and large the most significant factor in how touristy a place is anyway - on the lake, you will have avoided peak season with the larger masses, and chances are good that is also true for the mountains.
Generally speaking, the whole area is, while quite touristy price-wise not the kitschy type, as far as I remember.
Now, the alps around Innsbruck are quite high and (provided good weather) stunning - and close to Garmisch. How the compare to the high alps in Switzerland is something I cannot say, however.

I highly recommend you add Mainau to your list of places to visit. And if you go down the East side of the lake you can visit Meersburg en route to Lindau. Zurich is not in the mountains, so I'm not sure why you'd go there if mountain are what you want. You'll find more small craft shops on the West (Swiss) side of the lake than on the East (German) side, in my experience. You can hike in the mountains easily enough by taking the cable car up from Dornbirn on the South end of the lake. If you want to have dinner up on top and watch the sun go down (which is worthwhile) you'll need to make a reservation.

You are not going to miss the tourist season. Mid-September is still high season. Season ends mid-October. Switzerland, and everywhere around the lake, is expensive. More so as the dollar falls.

If you want to see some mountains and towns that don't get many tourists until the ski season hits, from the south end of the lake drive to Feldkirch and then take S16 towards Otztal. Another reasonable option would be to take a round trip on the Swiss rail system.

Be aware that driving either side of the lake in SEP can easily take 3-4 hours from north to south. It will be harvest time and the tractors will also be sharing the roads with the caravans, trucks, and tourists.

Great tips for around the lake. Otztal looks like a really good option for a mountain location as well. I think we’ve narrowed our mountain town choices to Otztal or Seefeld in Austria or Guarda in Switzerland.

Also, we’re just flying in and out of Zürich since it’s the closest international airport to Radolfzell. We’re not planning to spend much time there, just the afternoon before our flight home the next day.

I would say that your biggest obstacle is to find places to stay. We will be in Switzerland and Lake Constance during your time period. I made reservations quite a while ago. When I tried to adjust a couple of reservations by a day just a few days after I reserved there wasn’t a vacancy. Try seeing what’s available at the places you’re considering and plan by that. You don’t mention whether you will have a car, which will also make a difference.

Definitely seconding Mainau, in case you are even remotely interested in something like it - it‘s an island transformed into a giant botanical garden / flower park by a Swedish count exiled there in the fifties.
At the time you go there, it is possibly around the beginning of harvest time for both grapes and apples, so very young wine and cider will be available and are a local specialty - traditionally served with „onion cake“, very similar to quiche Lorraine.
Accomodation can definitely be tricky - it is no longer absolute peak (I maintain that stance) but still far from quaint or empty. A quick check showed still quite some vacancies, but it is definitely not on the cheap side yet.

I should clarify that it was in Switzerland (Berner Oberland & Lucerne) that I had trouble changing reservations and ended up having to change hotels instead.

Thanks everyone for the suggestions. We decided to spend 3 nights in Ötz, Austria for some hiking and then 3 nights in St Gallen for shopping and a side trip to Appenzell and hiking around Säntis. It wasn’t too difficult to find accommodations in both places at this late date and indeed in the more heavily visited areas in Switzerland, accommodations were a lot tougher and much more expensive. Also St Gallen lets us avoid staying in Zürich our last night since it’s only 1hr from the airport.