So, 1, to be secure, you need it to latch at the bottom really, not midway up the door. 2, To keep your hands out of the poop, it needs to be operated from midway up the door or higher.
My vote would be a slide latch installed vertically. I would go for one on each side but you could do one in the middle. On the bottom, on the tractor frame you would put a piece of pvc pipe, an inch or 2, oriented vertically for the rod to slide into. On the door i would use a wooden dowel aligned to go into that pipe installed with a couple of pieces of that same pipe, (like the picture in the earlier post of the gate slide), only vertically instead of horizontally. You could use a smaller piece of pvc for the rod, or a metal pin bent at the top into an L for a handle, but It's easier to put a handle on a wooden dowel than a piece of pvc, which will break quickly, and harder to bend a metal rod, unless you have one lying around.