I'm a little confused about the time you have available to spend where.
Zion is head and shoulders above any other park in Southern Utah, IMO. If you are time limited, go there first and do it right. Its close to a 3 day drive from Beaumont.
I assume that you have no interest in the Grand Canyon, for some reason. You will pass near it.
Yellowstone is awesome! Spend more than a day here to do it right.
Mt Rushmore is near Spearfish and Sturgis. It will take the better part of a day to get there from Yellowstone. Once there, you stand in a viewing area and look at it until you have seen enough and taken enough photos. It's a beautiful sight, but that was 10-15 minutes for me and others.
Some unsolicited advice. Once out there, traveling from Southern Utah to Yellowstone to Mt Rushmore will require alot of time on the road. I would suggest that you log onto
U.S. National Park Service - Experience Your America and look at the parks in areas that you are interested in, then decide what fits the time you have available. If you can't see everything, you have an excuse to make another trip. At least that is the way it works in my house.