I know there had been some fight ideas and i hope this one is new. I like the idea of there being safe zones and such (like Aroxy mentioned) but when it comes to actually fighting i think it would be fun to have a whole bunch of different moves available (different ones for felines and canines) that you could choose in battle (ex: bite, scratch, push over, etc).
Maybe when you are in the battle, you have a chance to push the dodge button and if you do it at the right time, you can dodge an attack, and maybe when you attack, you have to keep pushing a certain button to strengthen the attack. Also the size of your character could affect how strong you are.
The player should have a chance to run away if they want, and maybe when you are playing, you could have a chance to kill the other persons character, and if the character dies, the person has to restart from the main menu.
Maybe when a person successfully wins a battle, their character could get a couple scars where they were scratched or bitten (if they want). Of course, there shouldn't be too much blood involved.
Well these are just my ideas, thanks for reading and i hope you consider them