They hold matches similar to what you describe from time to time at the Ellisville Dirt Pit Shooting Club near Ellisville, MS. They also do other special matches including at least one annually with hunting-type or caliber handguns. Encoreman may have more info. I've only shot their regular match so I don't have much specific info from there.
In general, although it sounds like it could be fun and would probably be a good learning experience, one problem I foresee is defining the equipment rules. Whenever you put the word "match" into play, envelopes get pushed, if you know what I mean. While a 5 or 6 shot snubbie is probably the most common carry gun here, I bet they would be in a serious minority at a "match", and it would be pretty hard to compete with one against someone packing, say, a high cap auto IWB. When you start trying to define what is allowed and what is not, things get tough and feelings get hurt, somewhat limiting attendance I'd wager.
As a bit of an aside, Poplarville used to host night matches, and they were, quite literally, a blast. LOTS to be learned competing at night. Attendance waned though and they haven't been held in a while. I just said recently that it would be cool to do one or two a year at least. So, why not a night carry gun match? I mean, everyone has shot their high tech whiz bang self defense ammo and knows how bad the muzzle flash is in the dark, haven't they? (You might be surprised if you haven't). One problem with that is, almost no one actually practices, let alone competes, with their self defense ammo. "It is too expensive" is the most common excuse. It can also be embarassing or worse when you find out it either isn't reliable or completely flash-blinds you for several minutes when shot in the dark.
But who carries a full size and 8 mags everyday ?
You mean you don't? LOL. (I don't either).