Hi Charlie,
You say "lawn", which to me means you want a smooth cut. Depending on your budget and just how smooth you want it cut, there are several optons.
2 acres is not too big to mow with an ATV. There are several self-powered "pull-behind" mowers that work well. I would go for something at least 48" wide. Less than 48" will take awfully long to cut 2 acres.
Nothing cuts better than a gang reel mower, but there is a lot of blade sharpening! Last time I sharpened a reel, it took about 15 minutes for a light touch up. However, there may be 5-7 reels on a gang, and that would take a while! You would still need something more agile to trim around bushes with. A bonus that is often ignored is that most weeds can not survive repeated low cutting. Reel mowers are the champions at low cutting.
A rear finishing mower pulled and powered by your tractor is a great way to go, and costs about like an ATV pull-behind mower. Prices start at ~$1500 for a light duty one, and go on up to ~$3K.
Right now, I'm using my 6' brush hog as a finshing mower. I have it set to 2" height and keep the blades sharp. It does surprisingly well. You may want to give yours a try. Use it everywhere except your immediate yard, and you may be satisfied with the cut.
Ag tires are fine on your tractor until the ground gets soft. When it rains, you will either have to wait until the soil dries out thoroughly, or risk leaving some real ugly tracks.
There are "Turf" tires, which look sort of like huge ATV "wooly bugger" tires. These are real easy on your lawn. R4 "Industrial" tires are sort of in between ag tires and turf tires.