How about a rabbit or weasel? Or if you're looking for something more exotic a Civet or Tasmanian Devil?

Oh those are good actually! I think it's okay for a few exotic races, you know what I mean?

Well, choose your species based on the personalities you want represented.
Weasels and Ferrets can be the mischievous ones.
Rabbits and Frogs can be cowardly lil' comedians.
Lizards can be tough guys
Oh, and the bear is the cook.

I'm guessing Kaiju are out of the question?

You're doing a pirate story? As in pirates? That sounds fun. What about shark characters, maybe as villains? Or a parrot pirate. See, I offer you originality. Monkeys are an option, but sometimes they can look too close to people. You could always choose one with an insane face, like a gibbon or a baboon. There's always birds, like albatrosses (That has to be spelled wrong) or owls,peregrin falcons, vultures, sparrows... A cute mammal might be a kangaroo rat or a kangaroo. Badgers are always a good stand-by as well.

Hmm, let see... dogs, birds, cats and reptiles seem like a natural fit. And of course you can never go wrong with bats! XD

maybe a marsupial like a kangaroo , or a crow for the crow's nest

maybe a foxy crewman ,

I suggest an eagle as the watcher at corw´s nest, fox as a cremember, tiger as a captain or second in charge

Red pandas. When in doubt, it needs more red pandas, lol.

Still though, if there is some sort of asian-ish type of pirate, they fit well. Plus their cute factor is a awesome contrast, hee...

You have to have an uprighteous otter guy somewhere about if it involves pirates!

Ooh Pirates, I love pirates what about a badger?