There are two courses in Birmingham-Ross Bridge & Oxmoor Valley. Both are great courses, it's been a few years since I have played those but they are usually in great condition, and offer challenging shots with good views.  If you have the time, venture out to Capitol Hill, very challenging, no one has every regretted it.  It's like a 1-2 hour drive if memory serves me. Overall the RTJ Trail is great, you can't go wrong.  Every year me and my friends list it on our places to go play.

Try Ross bridge in Birmingham, great hotel on site. And venture south to Montgomery with the Capitol and the judge. Enjoy. Also ask the weeding party if they have a connection to play Mountain brook on the east side of Birmingham 

What part of town are you staying in? There are good public courses all around town. 

Play Ross Bridge.  My favorite on the Trail.  Oxmoor Valley, also part of the Trail has two courses and  is only a couple miles from Ross Bridge, so there are three courses to try in the Ross Bridge area.  The Ross Bridge resort is also a nice stay.