There aren't a lot of places where you can bring your own food, but my husband and I went to Hillcrest Hall (www.hillcresthallinlittlerock.com) and we really liked it. Centrally located, pretty, and you pretty much have the run of the place. I also have a wedding I am photographing this winter at An Enchanting Evening out near Roland (www.anenchantingevening.com). They allow you to bring in your own food and drink, which is one reason this couple chose to get married there.
One of my favorite venues in Little Rock that allows you to bring in your own food and drinks is The Women's City Club ( http://www.jllr.org/?nd=rentals ) in downtown Little Rock. It is located in the Junior League Building and is a stunning ballroom with fabulous views and gorgeous chandeliers. It is a wonderful blank canvas - wood floors, neutral walls - that allows you to decorate with any colors or theme you choose. And best of all it is within walking distance of all of the River Market hotels where your guests would most likely be staying.
The only catch is that they allow you in just 3 hours prior to the event to decorate, but if you have help - an event planner or family - it shouldn't be a problem.
The church I go to is on Napa Valley Drive and it is the Annunciation Greek Orthodox church. They rent to non members and you are allowed to bring your own food and drink and whomever you choose for catering has access to our industrial kitchen. You can see the specifics of the space at the following link: http://www.orthodoxchurch.com/rental/. Hope this helps.
I've also done weddings at the Annunciation Greek Orthodox Church and it is a beautiful ballroom! Gorgeous space and they allowed us to bring items in the day before to set up. They also allow alcohol to be served in the space, so no real restrictions.