Welcome to Justanswer! I am Dr Altman and happy to assist you both today!
It sounds like you did everything right by rinsing out her mouth and monitoring for any chemical burns to her mouth as well. As long as the battery was not ingested I would be feeding a bland diet for a few days as a precaution, white rice and boiled lean meat (chicken as an example) along with pepcid (famotidine) 0.25 mg/ pound every 12 hours x 3 days which is an over the counter antacid.
Since you did not see any redness in his mouth, I think he was fortunate enough to avoid a burn. The puppies I see with this problem have an obvious area of inflammation where contact occurred, and it is quite painful. They often have difficulty eating their regular food and are depressed.
I recommend monitoring the rest of the day for lethargy, reduced appetite or difficulty eating. If you see any of these, I would take her to the emergency hospital for an exam and pain medication.
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