Mike,
Have a little advice for what it's worth.
1. I've never tried a hand truck but would think the small wheel size would be be a real pain in the butt in rough terain, plowed fields, etc.
2. Don't even think about a single wheel cart unless you're going to have smooth straight running. You'll kill yourself trying to keep a 200 lb deer balanced over that wheel in rough going. Been there, done that with a friends cart.
3. I built a two wheel cart a number of years ago out of some scrap tubing and one of the boy's discarded 24" bikes and it has worked very well. But you can get nice light weight carts now days for around $100 and if you have to buy the materials to make one you'll probably spend more than that doing it youself.
However if you have access to some scrap tubing, wheels and can weld I could post some pictures of mine to give you some ideas. You might have to wait a few days though as first gun season starts this Saturday and I WILL be somewhat occupied.
Bob