How to ease indoor cats eye infection?
If the eye got scratched (either while catching the bat, or from the other cat), and infected, a topical ointment will treat it without passing it to the other cat. If the eye issue is due to a virus (chlamidya, congunctivitis, etc) the virus can certainly pass on to the other cat. Try to quarantine the cat in the bathroom, with a litter and food, that way you can also easily tell whether the cat is eating, pooping normally. Soak a facecloth with warm water and wring it out, start stroking the cat’s head with it, like you are grooming her, then gently around the face, dampening the area around the eye, then wipe the eye and go back to stroking the head. If you only focus on the eye, she may fight you on it, and it will get difficult to treat with ointment when you do go to the vet. Good luck!
Cats often harbor a virus in their eyes, which can flare up. My cat secreted a greenish pus like you described, and I immediately rushed her to the vet. It checked out as rather routine, and I was given some eye drops and it hasn’t flared up since. Using a warm wet cloth to wipe away the discharge is probably all you need until you can get your cat to the vet. It is unlikely to cause a problem for your other cat, although I am not a vet. The fact that your cat had all the shots shows how important that is, considering the danger the bat posed for your cat.