Agreed. But, ice or no ice, what water temperature to them is fatal?
As for the Mississippi, I don't believe it's tidal all the way up to the Tennessee line. Additionally, it flows north to south. And the major tributaries (like the Ohio & Missouri Rivers) flow through some very harsh winter states. So, if anything, I'd expect the Mississippi to run very cold through Tennessee -- even if Tennessee itself is having a mild winter. I would think the chill of the river would act as a repellent to southern reptiles.
Though a full zone colder here than Memphis, in all but the coldest winters, our creeks (of any decent size) and rivers do not completely ice over. They stay mostly liquid. I think complete freezeover is a Zone 6 thing. Point being, I'm afraid of gators coming here. Also afraid of cottonmouths and fire ants, but that's another story.