I agree, find another vet if you can. A vet that only wants to give 12 days worth of antibiotics for a URI in a rabbit, is NOT a good rabbit vet! Of course it's preferable to not have your rabbit on antibiotics for a long time as it can sometimes mess up their gut flora, but with some rabbit illnesses it is needed. URI's are notoriously difficult to clear up, can take weeks to months, and even then may not completely go away and will need to be retreated. And the problems with not treating it long enough is that the bacteria will develop a resistance to the antibiotic so it will no longer be effective, as well as the infection will continue to get worse and could progress to pneumonia, or possibly spread in other ways. So treating properly and aggressively, is important to do. A good rabbit vet would know this.
With seeing some improvement with the baytril, another round may help clear it up, but if not you may need to get a culture done and/or try a different antibiotic. Something that you will want to be aware of is that some antibiotics can cause a loss of appetite, which will also require syringe feeding if the antibiotic is continued. Usually though, it's better to try a different antibiotic if this happens.