I would highly recommend taking your chin to the vet before dosing her with Baytril. You are probably right that it is an upper respiratory problem, but only vets are legally allowed to prescribe, dose, and dispense drugs like Baytril.
Vets go to school and become licensed by the state so they can diagnose and treat animals. Before starting your animal on any drug, particularly antibiotics, a vet will consider many things like:
1) Is it the right drug? This is more complicated than it sounds, since it has to do with where the infection is as well as many other considerations such as the type of infection. A culture may be necessary to treat the infection and keep it from coming back, and once you dose with any kind of antibiotic, you change the flora and you are unlikely to ever get an accurate idea as to what is going on. Also, there are other things that can present similarly to an URI that your vet should know how to identify.
2) The amount of antibiotic necessary and the length of treatment. Successful treatment of any infection hinges on working with the body to destroy the infection at one go. A shortened round of antibiotics or an improper dose increases the likelihood that your chin will develop a resistant infection. The longer this battle is, the more likely your chin is to get worse or even die, so you want to get it right the first time. So starting a home treatment, and then finding it didn't work (or worse, it worked but then more severe URI symptoms come back in 2 weeks) and then going to the vet significantly reduces the chances that it will be successfully treated. Also, there are MANY different forms and doses of enrofloxacin (Baytril) out there because it is a widely used drug, and without knowing the concentration (mg/tablet), dose, and an accurate weight, it would be impossible to accurately dose your animal. It is dangerous to guess, because improper doses can do great harm (like permanent damage to the liver or kidney, or death if it is a high enough dose).
3) Interactions and complications. Although there are chinchilla-specific cautions associated with most antibiotics because of the chinchilla's type of GI tract, there are other considerations as well. Baytril is a very harsh drug, and may not be worth the toll it can take on the body. There may be a better drug that your vet will consider, as there are many options. Your vet may want to give you special instructions or provide you with a non-oral form of the drug to cut down on the impact on your chin's GI.
So for all these reasons, I STRONGLY suggest going to a vet and not blindly guessing at a proper dose of a reasonably dangerous medication provided by unverifiable internet sources. If you are worried about finances, or really want to use your own Baytril, go to the vet and explain that you have some already. Take the pills with you. They may be reluctant and want to prescribe their own medication, but often they will be willing to work with you.
