Here in MN bringing game out whole is part of the deal. In my area we have a lot of similar gated trails and a game cart works great there, but you won't be using it off trail in any significant capacity. Find a place to stash it in the underbrush when you depart the trail.
A few pointers - You may want to figure a way to extend the handles a bit so you can stand upright when you're wheeling your way home. Seems most carts are built by Chinamen, for Chinamen. Load the cart so it is front heavy, i.e. you have to lift upward on handle. Back heavy or even balanced is no good. Finally, bring plenty of straps/twine/rope... dead meat has a tendency to really want to shift around and you don't want to be constantly stopping to adjust it.
I have some doubts that the trail you're looking at is anywhere near as flat as the ones back here, and you may find packing is the better choice. At that point your cart just becomes dead weight.
