Those Guys, Otta-smack-u will spawn at the beginning of the battle, followed by Squire Rel, B.B.Bunny, and more Those Guys. These peons will be spawned regularly throughout battle. After a few seconds, Pigge Back will spawn. Rain D. will then spawn after another few seconds in together with Dagshund and the same few peons. As you're fighting all of the above enemies, 3 Shy Boys will then spawn, one after another. Beat them and Rain D. and the win is yours.
Have your meatshields protect against Otta-smack-u with Paris Cat, and continue to save up your Cat Wallet. Make sure to have strong meatshielding once Rain D. comes out, otherwise it'll attack your attackers. Even when Shy Boys come out, it shouldn't become an issue if you have a substantial number of meatshields and ranged attackers. Take along Anti-Red and Anti-Floating cats if necessary.