I run on ice that usually has no snow so it is sometimes real slippery getting around. A few years ago I made up some tire chains for my 650 arctic cat. The chains have worked beyond what I had expected and then some, great job.
I measured the tread face and about 4" down each sidewall and came up with 16" for each cross chain. Once that was all accounted for I shopped wally world for a $38 set of car tire chains of that width. With bolt cutters in hand I rolled the tire chains up over the tires, drove over them 1/2 way and snipped off the remaining extra length. I put them the way I wanted and bunji corded them up tight. Including the trip to the store and all, about an hour of time and the chains at $38.
They work great on ice for pulling a load, and have made the machine a real beast in deep snow. They really help running the ramps up in the trailer too. Oh, they do not hurt the tires or rub on any of the brake lines or other essential parts of the machine. This is my fourth winter on them.
Fisherman, it is what it is. Changing tire sizes will change your final gearing and you will really need that lower gearing on that smaller machine. You can spend money on tires and on suspension to raise up the clearance ...and then a new drive belt, and well, hold off with what you have and get a bigger machine when you can. I had a 300 polaris two wheel drive and was in the exact boat you are in. I held out for one winter, got the same $500 trade in that I gave for it and went up to the arctic cat 650 for another $3,500 Now I can run with the best of them on and off the ice. I am glad I did not sink any more hard earned money in to the smaller machine. Sweat it out and try some good fitting tire chains for less than $50.
For tires, believe it or not, Walmart's auto bays in select stores can order in any tire size for a lower price than most tire stores, and at a lower price that ATV stores as well. To answer your other question on what kind of tire, a new tire has softer tread rubber good for ice somewhat and an aggressive tread is best in snow. Best brand = brand new.