The only place I know for SURE that they do that is at Coco. For this reason, we will never patronize Coco again, so long as that is their policy. As far as other bars which do not allow you to bring a cooler if you're renting their chairs, there MAY be others on Orient, but I don't know. This year, the status of bars is sort of up in the air, with the 'renovation' of Orient Beach, due to start any time now. So places that didn't crack down on coolers before MAY start to do so now, because there will be many fewer places to eat and drink, until they finish with the work.
On the other hand, PERHAPS if you bring your own chairs, you might be able to set up shop in front of the area under renovation. It is also possible that the places involved may rent chairs still. If so, presumably, since they can't provide any food or drink, they won't mind you bringing your own there. I think no one knows what will happen, until it does.