Unfortunately, if your home gets above 75F or so, there's really no way around the fact that an air conditioner is a life-saving necessity for chins. I know it's a considerable expense, but it's just like making sure you have a safe, properly-sized cage and the right kind of food. It's a necessary investment to give them the care they need to survive.
The cheapest option is probably a small window unit, followed by those portable units that look like plastic boxes on wheels. Those tend to be pretty energy-efficient, so they shouldn't jack up your power bill too much if you only turn them on when necessary. I have one of the portable units (my chin's in the living room, so the room is too large to be cooled by a cheaper window unit) and I've found it to be easy to install and use, and easy to uninstall and store away when I don't need it. I'm fortunate that the room where I keep my chin stays unusually cool even when it's hot outside (recently I panicked a bit when it got into the mid 90s while I was at work, and I hadn't turned on the AC that morning because it wasn't predicted to get that warm, but when I got home, I fortunately found that it hadn't gotten above 72F in the room where my chin lives). Even so, I'm glad I have the AC, because when summer hits and it's not cooling off as much at night, even that room will likely get too warm for her without AC.
If your chin is in a smaller room like a bedroom, a window unit would probably be the most cost-effective option. They're cheaper than the portables and don't take up as much space. If Chinzia is in larger room like a living room, then a portable is probably your best bet. They're a little more bulky and expensive, but they're usually more powerful than window units. They're also easier to install and uninstall, so you could store it in a closet or something in the winter if you didn't want it taking up space when you're not using it.