Cows on teathers work great. I have a yearling heiffer on a teather in my front
yard as I type this. Definitely dont try goats or sheeps with ropes, they have an instinctual death wish. Wont end well.
So long as you move the teather frequently enough, you would not need any supplemental feed. Just kelp and salt for minerals, and a tub of
water.
Plantain is excellent
cattle feed, much superior to grass, IMHO. Your animal will graze roughly based on body weight, so for your size pasture, one animal unit would work well. I would get a yearling steer (800 lbs or so), otherwise you would need two young ones to get the job done. For me, two animals on teathers starts to feel like a job, one animal is great.
Dairy cow on a teather is a beautiful thing so long as you keep moving her to fresh grass regularly, like three times a day I would reccomend. Dont underestimate how awesome a
milk cow is! But definitely not as simple as a
beef steer.
Do be aware that if you dont move frequently enough, you will trample your pasture into mud if it is at all moist, which sounds like an issue in your area. I know I sound like a broken record, but frequent moves is key with animals on teathers. good luck!