Yellowstone in 3 days?
 

Can a 4 year-old survive the drive from SLC to Yellowstone or am I forcing the issue?

and which properties should we think about looking into? (middle of september)

Mine could, but she was a good traveler. Every kid is different.

Property in Yellowstone?
I would stay in different areas(even thought this involves repacking car every day or two.

2 days in Old Faithful Inn
1-2 Days Roosevelt
1-2 Days Lake Hotel or Canyon

2 days in Grand Tetons. Jackson Lodge, Signal Mountain, Colter Village, or Jenny Lake

I don't see why not. I've been there, done that and don't remember it being an issue at all. There are plenty of places to stop for a break and I would certainly recommend that.

It has been too many years since I traveled with young ones, so I can't help you there. But as for where to stay, you need to get busy on that asap. In park lodging can be hard to come by this close to your arrival. But don't despair if they show no vacancies. People and tour operators book way ahead then release rooms they don't need before they have to pay a cancellation penalty. I've found the best way to secure a spot is to call rather than using their website.