I've never been to Courmayeur; however, London-based Fraser Wilkin from Snow-Wise says: "I can suggest the Hotel Berthod for a nice central 3-star B&B or Hotel Edelweiss for something really simple (2-star) but still very well run and charming in its own way. If they want something more upscale but good value, then the 4-star Hotel Cresta e Duc."
I skied there one day as part of a Chamonix trip. Stayed in Airbnb in Chamonix. Drove thru the tunnel in the morning, it was quick. I didn't know this at the time, but Courmayeur is like a fancy type place - I think it's like the Aspen of that region. I did however have a great day skiing...I was with my wife and there was a lot of great intermediate terrain with some fun more challenging stuff to ski where she could just wait on the groomed runs at the bottom and watch me come down. On mountain "Refugio" food was awesome. Trip report I did which included pics, it was in February 2018.
Don't have much advice on lodging, aside from I recall Airbnb having many choices, but seems you checked that. I have a feeling it would be expensive for lodging in that area, but maybe look more into the Aosta valley a couple towns over and maybe find something cheaper?
I've looked at places on AirBnb in Italy, up to 8 km from Courmayeur. Looks like July is high season and everything is booked. I'd prefer to be in Italy partly to avoid renting a car, and partly because I speak a bit of Italian now that I can build on. French is non existent.