We are planning a trip to Yellowstone in August with husband and two kids. I'd prefer not to spend a ton of time in car but we were late in booking so I am only able to get one night in the park. Plus, I think my daughters might like the pool at the West Yellowstone spot So my question, is how does this breakdown look?

We will fly into Jackson Hole and stay there for 3 nights. From there, we planned to stay two nights in West Yellowstone, one night at Snow Lodge, and two nights in Gardiner before heading back to Jackson.

The northern part of the park (the thermal features at Mammoth) could probably be cut to one night in Gardiner, since it doesn't really take that long to tour, unless you are including Lamar Valley to the east (which should be planned for a drive through for the wildlife). Going back to Gardiner for a second night would involve backtracking, and a longer drive to Jackson the next day.
You could do your West Yellowstone place for 3 nights (my kids also loved a pool at the end of a long, hot day). We liked the amenities, location, and price of Worldmark (Wyndham) (tho we booked through VRBO).

.>>We liked the amenities, location, and price of Worldmark (Wyndham) (tho we booked through VRBO).

Note that park lodging availability changes daily, as people 's original plans change they release rooms they reserved. I was able to piece together a weeks worth of lodging pretty late by checking often.

You could keep the 2 nights in Gardiner and do a rafting trip along the Gardner River (yes, the town and river are spelled differently - I don't know why)