A window AC will work just the same depending on the ratings. Portable AC units are more expensive and take up more space but they can be installed in places that window ACs cannot and can be installed in places that have restrictions on window AC units. They can also be adapted to casement windows quite easily. So there are pros and cons. I have a free-standing unit and I like it a lot. If you get one, either a window type or portable type, that is the right size for the room, and that the room and doorway are properly insulated, you won't have to worry about central AC at all for your pet and can run the rest of the house at a higher temperature. That is what I do in near-tropical climate near Atlanta, GA. There are days though when the portable unit cannot keep up but the central AC makes it liveable when combined. And, yes, it is expensive. It's one of the drawbacks of having chinchillas and birds as pets. But we also have a 3 year old who shouldn't be exposed to high temps either so it's not all for naught. Another thing we do is cook outside as much as possible in the summer using crock-pots and barbecue grills to keep the house temps down.
Weather stripping is usually a good start when you're looking for places hot air is getting in. Expandable polyfoam sealants can also be helpful if you find large gaps. One thing that does work, although it's quite ugly, is Styrofoam insulation with reflective mylar backing. This can be put in windows in rooms that are excessively hot. Another thing that some people do is install exterior turbines for the attic space. That was popular when we lived in Phoenix, AZ where I didn't keep a chinchilla, but we did that and it helped. That and lots of ceiling fans to keep the cold air from the central AC flowing through the house.
