Sorry to hear about your tooth

Yes, that would be an acceptable use for an EF. And then after you refill it(or while), start a medical sinking fund for such emergencies
in the future.

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Definitely sounds like an emergency to me! It’s medical and if you don’t take care of it you’ll be at risk for worse problems.
Me, I’d pay and then concentrate on refilling the emergency fund, because that’s simplest.