Since we can use rifles legally here in FL, I have my own personal idea of the ideal turkey gun. It would be a Savage 24 0/U in either .22 Hornet, .222, or .223 over 12 ga with screw-in choke. I have seen only a handful of examples of this gun and they are VERY expensive. Nevertheless, I'm going to keep looking until I find one at an affordable price.
Having only one shotgun shot is not a big deal. Most turkey encounters don't require a quick follow-up shot. The big advantage this gun would give me is the chance to reach out to that gobbler that hands up at the 100 yds range and refuses to come any closer.
Also, when turkey hunting here you never know when a hog or coyote will step out. Having the centerfire rifle there allows you to take them.