Try the Peruvian
Or The Cliff Lodge at Snowbird
They're polar opposites, Peruvian being rustic and old time lodgey, The Cliff modern and sleek. You can probably get adjoining rooms or a suite and hit your $. I've stayed at both, wife prefers the Cliff, I like the Peruvian for it's traditional charm. There are condos, but not a huge choice in Little Cottonwood.. I'd avoid Park City during Sundance. The ski hill is uncrowded, but you can't eat, get a cab, park or even walk on Main Street. Most of us townies stock up on food and stay home.
You might like Solitude in Big Cottonwood too. It's a very family friendly mountain, unpretentious, and has a charming base village, small and elegant.(some Beaver Creek people were involved in the design) The canyon itself, just a couple miles from SLC, is entered through a narrow, winding road with thousand foot cliffs on either side. Once you're in the canyon, it opens up to an undeveloped mountain wilderness with a few cabins, a store or two, and the two resorts, Solitude and Brighton, at the end of the road. It's really quite stunning, but so is Little Cottonwood- LCC is narrower and steeper. A condo is the way to go at Solitude, they're all good and right at the base. Creekside
is the best Condo, and closest to the lifts. It should be in your price range. Look for packages that include lift tickets.
Alta and Solitude are best for all abilities, the Bird is pretty gnarly. All are close together, and there is bus service between them, but I recommend a car so you can take some side trips and see a bit of the Wasatch.
Snowbasin and Pow Mow up north are good too. Remote and undeveloped, but that can be great in this world...will need car there. I don't know much about lodging, for us, that's always a one day seek and destroy powder mission.