I always get a chuckle out of the Gateway Geyser. It is the largest fountain in the world. Takes about 10 minutes at noon, 3 and 6, during the summer, in the Park across from the arc. The fountain is quite impressive, and has a 3 story elevated viewing platform bow and STL skyline. It only takes a half an hour to get everything you need to get out of it.
Cahokia mounds are another interesting thing East of the river. They are the largest original earthworks in America, and it’s quite a decent free Museum explaining what we know Cahokian people and their civilization. Drive from the town center and hiking to the top of the monks mound and all you’re going to get out of the Museum is probably a total of an hour and a half. If anyone in your group in to Native American things, that’s it, and a good and free.
All museums in Forest Park are good for short trips, because they do not charge admission, can only fill in the time you have and go away, you don’t feel like you have to reimburse the $60 investment in parking/entrance.
Soulard market is also a good hour or so, buy some meat products, pet chickens, drinking a huge cocktail before noon.
The architecture in the city is also quite nice and worth making sure that between the places, particularly some of the things in Washington close to the Museum and the Soulard and some nice stately homes on Lindell toward Forest Park.
As far as the City Museum, I recommend going later in the evening. The audience gets older and thins out of six hours, so as long as the child can stick together up to 10 hours, or later, it’s a better, calmer, less crowded experience. Check the schedule for circus performances, tumbling and aerial acts were amazing, and worth to have time, but they can also be skipped in my opinion.