Where should we go? Courmayer or Innsbruck?

Not Courmayeur. It's a nice town, but quite small for 10 days. The hiking is good but very strenuous, and the valley is not the best due to the Mont Blanc tunnel traffic. And the town is basically a dead-end, since it is expensive to casually cross the tunnel to Chamonix (you could do so once, but probably not twice).
For Italy, maybe Cortina instead: heart of the beautiful Dolomites!

Innsbruck would be better than Courmayeur for me but it is a city and you do have to "commute" in and out for hikes and all.

Cormayeur is a small village (less than 3000 dwellers), deep at the foot of the highest mountain in western Europe. Mountains are among the most beautiful but hikes tend to be on the strenous side.
Innsbruck is a medium size city home to 130,000, with a large university (the medical department is very good), some historic monuments, at the crossroad of important communications. A network of cable cars grants access directly from the very center to the top of the north mountain. As for hikes, it cannot compete with Cormayeur, but on a rainy day it has all the amenities of a city while you would feel stuck in Cormayeur.
Personally, I choose Innsbruck for my holidays since 30 years. Mittenwald, Garmisch and a couple of Ludwig castles are easily accessed by Innsbruck (easier with a car); Lichtenstein is not so close and it is not very interesting anyway. Even dolomites are a possible car trip, Ortisei being 75 minutes driving from Innsbruck. Salzburg and Munich are possible day trips (a bit on the long side).

So i have never been to Cormayeur, but in 2003 (long ago) my husband and I took a once in a lifetime 5 week trek around Europe for our honeymoon, with Innsbruck being our 2nd stop (1st was a 7 day sailboat cruise of the greek isles) so that we could "ski" in the alps. it was june....It was amazing! it had great food, was small and walkable; while we had to take a local bus to Stubai glacier, there was a great gondola ride, lots of hiking trails, and amazing little villages along the way, that we wish we had time to stop at. Salzburg is amazing too, if you could get there. I know it was long ago, but we loved it and in the early 2000's it was a great place for inventive, interesting cuisine. I am hopeful that it has only gotten better! Whatever you decide, have a wonderful vacation!

I grew up going to Innsbruck from our home in Unterfranken/Bayern (early '00, as late as '07), but I have no idea what it looks like post-COVID. You'd definitely want a rental car. It is very close to Garmisch-Partenkirchen, and from there you could take day trips to Neuschwanstein, the Romantische Strasse, and more. There is a stunning hike around Lake Eibsee, sometimes with icecream shops (might still be open in Sept) for a post-hike kugel. Innsbruck boasts the childhood home of Mozart. It's also a great place to get a plate of mixed grill with spaetzle or potatoes. We had booked a trip to Garmisch in May of 2020 (with plans to visit Innsbruck from that homebase). I love Germany and always have. It's a great place to visit. With that said, we've booked our replacement trip to Italy. It is magical in a different way.

Instead of Couermayeur, check out Cogne. Still in the beautiful Val D’Aosta, but a bit further off the main track. Absolutely charming place out of ski season, which I wouldn’t say about CM. We had such an absolute amazing B&B experience at Les Trompeurs/Chez Odette, that we asked if we could just buy the room permanently and live there.

Couple of sites about it:

https://www.e-borghi.com/en/village/Aosta/528/cogne

https://www.italymagazine.com/cogne

Here’s Chez Odette:
https://www.lestrompeurs.it/index.php?l=en

Don’t believe in an afterlife, but if I did, I’d expect it to look a lot like the entrance to Gran Paradiso that’s essentially the back door of Cogne.