Yes, pneumatics! I’m in GA. The last barn i was in, even with my cheap bad wire spoked pneumatics i had to get off and push from behind the cart to help my pony get through some spots. Thank goodness he’s a good boy. Now i dont have pneumatics at my new barn, but my trails, though sandy soil, are covered with pine needles and with the exception of a few spots, my wheels do not sink.
If you need to keep it cheaper and dont care about looks, you can buy a better quality metal easy entry cart and put new wheels on them. I’ve got a thread on here somewhere about how my wire spoked wheels crushed and then my cart fell apart… So dont go wire spokes! You just never know what might happen!
I know some people have gotten motorcycle tires and put them on, you might have to put a new axle on the cart to do so if they do not line up.
Most all marathon carriages and wagonettes can come with optional pneumatic wheels if bought new (or found used) you are just looking at a possible jump in price depending on what you are looking to spend on a new cart. (I know the wagonettes at the auctions sell new for roughly 2-3k right now, a bit of a drive for you, but the Southern GA Carriage auction in Perry is coming up in May)
You could also contact some of the larger carriage dealers in the US and find out about ordering new wheels for whatever cart you end up with. They wouldnt come large enough to work on your meadowbrook i dont think.
Google:
Summit View Carriages
carriage driving essentials
driving essentials
horsecarriages.com (which is the only one i can think of off hand that actually offers a few training carts with pneumatic wheels)