If you just want to stick to the popular commercially guided peaks then I'd throw in Mera Peak (Oct-Nov).
I've not been on Ama Dablam but I'd say it's more Oct-Nov and into Dec than Sept. Crowding dictates some of that.
Ditto Cholatse is more of a Oct-Nov peak, than Sept. Probably Bartuntse too.
Lhotse in post-monsoon? You're going to need to pay Sherpas to fix the icefall for you, just for Lhotse, without the economy of the Everest crowds, so that's unlikely for most.
Spantik might be fine in Sept, but it's more popular June-August.
On bigger, harder peaks like G1, the summit days are often concentrated into just a few days all season, due to weather and teams' itineraries. Most approach in June, to summit later in July or early August. Aiming to summit late August might be OK for weather, but you might be the only team trying. That can be good or bad.
Nanda Devi has been climbed in various months, but usually May-June and Sept-Oct. It is off-limits anyway, for environmental reasons, and has hardly been climbed for many years. If you want a doable big peak in India, consider Satopanth.