People in Liberty suits have rights, too, but not the unrestricted right to solicit customers from a median. While this does implicate the First Amendment, it would be the kind of time, place, and manner restriction that usually passes muster. The situation would be different if a local government tried to completely ban the use of such "moving signs" or (as I prefer to call them) "business mascots," which of course is something that has happened before. See "The McHenry Code," Lowering the Bar (Sept. 6, 2006).
b Coincidentally, that incident (which happened in Illinois) also involved "Lady Liberty," as well as the Verlo Mattress Factory's "Mattress Man," a 4-by-3-foot ambulatory mattress with "comically large hands." McHenry's city council had decided that such "live moving signs" were distracting drivers (which is part of the point of having one) and causing a nuisance because people honk at them. (The council also threw in an alleged "safety risk" to the person in the costume, saying they might get heatstroke.) If I recall correctly, the council later reversed itself on the complete ban, thus giving Liberty some limited freedom.