I wouldn't take out a loan for something that small. Better plan is to find a way to deal with your carpet, and keep it clean, until you can save the money to install the tile. Get some side jobs and pay with cash.
Also, you might save some money on install if you find someone at your church or a friend of a friend who would be willing to give you a deal, rather than pay full price for an established business. On the other hand, you don't want to hurt your home's value (I assume you own this residence) by a sub par install job.
Have you thought about wood laminate flooring? There are some varieties that are very scratch resistant and good for animals. Might be cheaper than installing tile, especially for that large of an area.