I am not a professional. The square brine tank takes less space, and it would do the job nicely. Yes, a round one is structurally stronger, but the square ones hold up for almost all people. But then I figure most people don't cram them full.
A softener with 1 cubic ft of resin is big enough for you. That could be done with a 9" x 48" tank. I would go up to a 10" x 54" 1.50 cu. ft. tank myself. It takes only a little more floor space. I would lean toward the Fleck 5600SXT if I was doing it myself. Fleck restricts how you can buy their newer stuff.
In Tampa, some people might put the brine tank, or even the whole softener outdoors if protected from rain.
That Autotrol 268-760 unit was revised about 2001. Still, I don't know what feature would be better for you than a Fleck 5600SXT.
A cabinet softener is not a bad choice. Few professionals go that way. While cabinet units tend to not get repaired, they are usually good for 10 years or more. Then you buy a new one if needed. They take less floor space.
On your water test results, presume your chlorinated water has zero iron. That test result you posted was probably from before the water got treated. But because your water is chlorinated, opt for 10% crosslinked resin if it is offered at a reasonable adder. And until you know otherwise, I would set up your softener for 15 grains of hardness.