Anything more you remember about this game? The setting, perhaps? Was the action happening in a city? Was it a sidescroller, multi-screen or isometric (like Vendetta). One game that comes to my mind when I hear about peeing is Wild Streets (1990), but that's only because of kick animations of both: player's and enemies' that, as someone has written on Lemon64 forums, remind peeing dogs.
You may want to browse through the list of C64 beatem'up games. Check this out:
http://www.mobygames.com/browse/game...ng/action/c64/