It sounds to me like an upper respiratory infection (URI). I would NOT give your cat Benadryl or anything of the like. I would highly recommend seeing your local veterinarian who can prescribe the proper antibiotics/medications and the correct dosage for your cat.
I would be very cautious in...
Keep in mind that bats can also carry rabies and it can be transmitted in their feces. I do agree that it can weaken their immune systems, and if it is not required by law in your state then an older dog could be vaccinated for rabies every 3 years.