There is a very good chance that Bridgit has FHV infection. It is extremely common.
The antibiotic won't help it since antibiotics only affect bacteria, and not viruses. That said, some of these cats do get secondary bacterial infections. But most are simple and straightforward kitty "colds".
The most important thing to do is keep her eating - good nutrition gives her body the energy to fight it.
FHV is, like other herpesviruses, not curable. It will be with her forever, but may or may not ever cause trouble in the future. I always put these kitties on a lysine supplement which serves to make it difficult for the virus to replicate, and helps the cat get the upper hand.
Lysine is not a drug - it's an amino acid - so this is a nutritional approach with a good track record. FHV needs arginine to reproduce and the lysine blocks arginine receptor sites.
I use Viralys powder, which is available OTC online and through larger pet stores and at many vet clinics. I give my patients 1 scoop twice a day for at least 2 weeks, then eventually decrease it to once a day for long-term maintenance.
Some cats just need to stay on it or they keep having relapses. Many do not, so it is used temporarily. There are no side effects. You do need to mix it into a tablespoon of canned food.