Yes, as long as there is no rule against it (and there isn't) go ahead with your EE cart. You asked about single pleasure, country pleasure, and classic pleasure, but there IS a rule that you can't cross enter the same horse in the same show in any two of those divisions. At a local show, where there usually isn't a lot of time between classes, some people wear the same - or similar - clothes to what they wear for halter. I have never seen gowns for local one-day shows around here.
Do you have a Pinto? Because at the New England pinto shows, you see a wide variety of carts and carriages and your EE would fit right in!