Hi Dean..I just shared your exact same experience for the first time yesterday! In Michigan as well, and yes...in the snow lol.
Honestly, anything will work but you probably ain't finding your arrow with a field point. I shot two yesterday with a judo point and a Saunders bludgeon tip - the ones that look like little meat hammers made out of plastic.
The Saunders bashes and breaks every bone it makes contact with. The judo will go through the bunny and tear it up. They both work but you are right - in this snow, you will probably lose an arrow. IMO, a really heavy arrow works great in the quiver for a cover shot when they hide!
Also...another option is to put a steel washer behind a blunt or rounded field point. 85 cents and they are great!