I will be completely honest. It is Very very difficult to rent the "park" and is also very expensive. It is managed seperately from the Arch grounds. 9 times out of 10 they only allow large groups to rent the grounds. This could of changed but I believe it is not marketed or targeted to Weddings. Have you looked in to Forest park? They have many outdoor ceremony venues. Alix-Ann Wedding planner
There are so many outdoor venues from as simple as Stacy Park in Olivette to as complicated yet beautiful as the Studio in St. Albans. Try all the venues in Forest Park, the World's Fair Pavilion or the Jewel Box.
You might want to check out The Muny in Forest Park which has two pavilions that are private, have a roof, fans, and electricity, but are open and give the outdoor experience without the hassles of tents, muddy lawn, etc. Can have a reception there too and use your own caterer which can save lots of money.
I have recommended The Muny many times. I would not say that they are inexpensive. The rental fee does not include tables and chairs. Like most outdoor locations - those things must be rented. Having your own catering company does not mean it will be less expensive. It means you can have a choice in your menu and other options.