Avignon to Geneva- Place to Stop Along the Way prior to flight out of Geneva Airport

You will need to be at airport by 9:45 am
I would just stay in Geneva to avoid any risk

Lyon? Has train to Geneva every two hours, takes two hours"
https://www.seat61.com/international-trains/other-cities/trains-from-Lyon.htm#Lyon_to_Switzerland

Have you had a look at Google maps to see the train times available? You are correct that you wouldn't have time to travel from Avignon on the morning of departure. Keep in mind that you need to be at the Geneva airport by 0945 for your flight. Lyon might be an option if you took the 0638 train. But at a distance of 160 km, you would be hard pressed to come up with a plan B if something unforseen happened en route that morning. Spending the night before your flight in Geneva may be less risky than having to pay for a last minute new ticket because your train was late or cancelled. That doesn't mean you have to spend the day before in Geneva. Just arrive on a late evening train.

Maybe Evian, France? If you stay for two nights you could enjoy the spa town of Evian (yes, Evian water) and maybe one day take a train to Montreux or take a ferry over to Lausanne. Look at Swiss rail for information on the timing of a train/bus to the airport from Evian. My first glance at SBB looked like it might be doable. But, of course, the safe choice is staying in Geneva before your flight.

Annecy is an obvious choice. There are buses (don't even need a train) that go to
Geneva Airport and take 45-60 minutes.

If you're willing to go past Geneva and then backtrack, Lausanne or Montreux are
other possibilities. It's about 45-50 mins on the train to GVA from Lausanne (going thru
Cornavin) and 1:15 from Montreux.

I agree with shoeflyer. We spent our last 3 nights in Annecy and booked a driver to get us to the airport in Geneva. It was an easy 45 min drive. Annecy is a lovely town on a crystal clear lake at the foothills of the French Alps. On the other hand, Lausanne or Montreux are equally lovely. Also an easy commute to the airport. Head on from Avignon, get closer to the airport, and enjoy a lakeside town!

Annecy is about the only choice IMO, or Geneva itself, or small towns on the lake like Yvoire (arranging a cab). Bad traffic from Annecy to Geneva in the morning though, so have a good buffer if taking the bus.
Lyon is too far away.

hey hey On the road!
no way would i leave avignon to geneva on day of flight. never know what can happen on the train. we always at departure airport night before international flight. when is this trip?
we spent 2 nights in geneva, not my favorite, and a 40 minute taxi ride to annecy for 5 nights. loved annecy on the lake. lots of things to do and see, many restaurants to chose from
ovonetwork.com/ annecy street market
frenchmoments.eu/ old town revealed
in old town on tuesday, friday, sundays 7am-1pm. it was busy on our saturday roaming around the beautiful canals, flowers abound, mountain views, busy vendors of everything. stopped at art store and bought small handmade framed painting, had a yummy fondue prix fixed lunch down a small alleyway near river thiou outside table (la rotisserie du thiou)
eat crepes, croissants, bread & other pastries
we also stopped at italian restaurant in city center to have rose wine, calzones, pizza. walk part of lake or that a boat ride.
annecycitytour.fr
imperial palace annecy a beautiful hotel right on the lake.
since your time is short enjoy this small lake town. look at train or bus schedule, the station is close by and it's your decision getting to the airport before your flight. have a great holiday and enjoy
aloha