Your way of cleaning rabbits is pretty close as to how I clean squirrels.
Usually the temps are pretty cold when hunting squirrels and rabbits, so the need to field clean is unnecessary. Never had bad meat from these "home cleaned" animals.
Snowshoe hares can be a load so our sons who were too young to hunt wore backpacks and were the "game carriers". When their loads became too heavy, they went back to the truck and unloaded....couldn't get lost seeing they just followed the tracks in the snow.
I prefer cottontails for eating because snowshoes are a little stronger tasting and tougher, but still made good meals.
There's no bag limit on snowshoes in Wisconsin, so when they're on the top of the cycle, 3 of us killed at least a couple dozen many times in a morning's hunt. Early on, .22s were only used, but then I started to use a .45 PRB flintlock and had no problem shooting as many as the guys using the .22s. We mostly got head shots.
At the top of the cycle, there's mostly young hares which aren't very wary and probably never saw a human....close shots were very common. Besides, these young hares were much better table fare than the older ones......Fred