So sorry to hear about your dogs health issues and I am so glad that you are looking at making a special trip. Unfortunately, I live and travel mostly in the dark cold of Alaska but I just wanted to wish you luck finding a spot for you and your friend. Best of luck.
Our canine companions live to swim as well and we always look for places for them to enjoy the water when we are out traveling.
Your listed requirements naturally rule out many locations, however, along the Gulf coast there are some places you can try. This past November we stayed at Gulf Shores AL and the state park at Fort Morgan has a pet friendly beach. We had it to ourselves. It is only about 100 yards from the parking lot. There is no campground at the park but several are nearby.
There is also a dog park with a small lake just down the road from Ft. McAllister SP south of Savannah GA. It is a community park, but, again we had it to ourselves.
We go to South Padre Island in TX every winter, mostly because the dog loves it! Ok, we do too.
Campground on the south end (Isla Blanca County Park). We then go to the north end and drive up the beach far enough to let her run and swim with no one else around. If you are adventurous, you can camp right on the beach.