See the attached photo. That’s a hot dog sausage(1), which had a CR2032 inserted into it for 3 hours, though noticeable damage was already present after the first hour. That is due to electric current alone: no leak from the battery and no heat damage (everything was sitting below 22°C in 24°C ambient).
Well, there is very little damage in the esophagus because it spends less than a second in it, before sliding into the stomach.
In the stomach, the damage is repaired within a couple of days, because the lining is continuously renewed. (The acid in it does much, much worse damage to a hot dog sausage, too.

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Within the intestines, the damage is spread out over the entire length, and there is a combined mucus and bacterial covering that will take the majority of the damage.
So, although I agree with the damage, human (mammal) digestive systems are quite robust, and deal with damage (as long as the item moves along smoothly). Having the CR2032 lodged in would be very dangerous, though!
In the doggo's case, the charge in the cap is of course neglible, and would have been discharged in the stomach (where any slight damage would be completely irrelevant due to the lining continuously renewing due to the very acidic conditions). (If it had discharged in the doggos mouth, we'd know.)
Chemical toxicity and having the cap lodge in somewhere were my main worries; but since it turned out quickly and intact (with almost the original capacitance, indicating there couldn't have much loss of electrolyte or structural integrity in my opinion), I'm happy that the doggo will be absolutely OK.