What unit or units would be good choices for measuring the weight of the ring in the bull's nose?

Answer: Well, ..., since you asked for weight, it would be Newtons = kg m/s^2, or better, dynes = g cm/s^2. Or ounces of force. If your bull is in Denver, the weight would be a little less than at sea level. Move to Denver, lose weight!

But actually, assuming you can get it out of the bull's nose, the best unit would be whatever your scale is calibrated in. Or tenths of ounces, or grams.