My one and half old cat dragged into my house (kitchen) a half alive/dead bat. I didn't notice when I let her in, but then when I walked back into the kitchen noticed the bat on the floor. This was in the morning too, which is obviously a sign something is wrong with the bat. The bat didn't have any blood on it or anything just was laying and flapping a bit on the ground. I put a plastic bowl over the bat and used cardboard to maneuver under the bat and bowl. Then I took it outside to a field and threw it free. Threw out the cardboard and bowl obviously and called the vet. The vet said the cat would be fine since she had her rabies shot. And my cat was perfectly okay and has been ever since. I didn't need any shots myself and I just washed my kitchen floor with bleach and some other products.
My cat is serious hunter! She also caught a bird by its wing when it was flying away and off the ground too. Luckily I was able to distract her so she let go of the wing and boy did she give me the nastiest look ever! If your puppy has had it shots he should be fine and sounds like you took safe steps to get rid of the bat too. Hopefully everything is okay.