Stall the Doges and Snaches with Wall Cat to generate money. You can increase your chance of winning by maxing out your worker cat to level 8, or generate money at a lower level and max out that current level's wallet.
When Mooth appears, repeat sending out Dragon, Sexy Legs, Wall, and Macho. Once you defeat the Mooth, some Doge Darks will be still alive. Defeat them and your units will start to attack the base.