With Forest Park you will get a recently refurbished muny, so the conditions are vastly improved from 5 years ago. The course is short but is fun and has some good short par 4s, which offer a variety of options.
Gateway is a very convenient course which is a very good value and a quality true public course. Annbriar has the more interesting layout, although is a bit further and does not always allow walking (and some may consider it to be a hard course to walk).
There also is an almost 100 year old design, Normandy. The conditions are not very good, and it is also short, but is a great layout. It is cheap, so if you do not mind the conditioning, you will play a quality layout and a layout little changed from its original design.
If you want to drive further, in Alton is Spencer T. Olin which has had 2 recent Publinks championships (men’s and women’s).
About 20 minutes south of downtown is a Hale Irwin design, which is not as good as Gateway or Annbrier, but is a quality course, Quail Creek. This course can also be quite a challenge.
Finally, if you can get access, St. Louis Country Club is a legitimate top 100 course in the US. Although the club has a reputation for being very limited in who it allows as a member, access is not quite so difficult, although will probably require some effort. But is may be easier than Bellerive or Old Warson, as anyone entertaining client in St. Louis during the NCAA Regional may be trying to get access those courses first.