I'd say yea you probably could get fired for it. if you didn't do it cleanly. that would include making sure you're not representing UPS in any way while doing so, like wearing your uniform or doing it in the parking lot of UPS or while your uniformed coworkers are spectating or boasting to a camera after bloodying your opponent and incidentally mentioning that you work for UPS. you'd have to make sure both parties to the fight consented to it. and that it didn't result in any significant injuries that impaired one's ability to work. if there was an injury, that did prevent you from working, misrepresenting the cause on something like a workman's comp form would be criminal fraud and the cardinal infraction of dishonesty. like if the other guy dislocates his knee just from footwork or slips and cracks his skull open, you're opening yourself up to all sorts of legal liability.
can you do all that reliably? and you're sure the other guy isn't going to betray you and take out a restraining order against you? yea the union would put up resistance, fight for you, but they're not going to do so with passion because you're acting like a jackass and making them look like a bunch of jackasses, they might very well let the company beat them at it.
Just don't do it. you're not in junior high anymore, stop acting like it.