I honestly don't see the "issue" being the tooth vs. birthday present.
Couples often disagree.
The idea is to be able to come to a peaceful agreement and have both people leave feeling "heard" and being able to honor the decision.
The fact that you guys can't resolve it with out you going into a tail spin seems to be the problem. I'm not saying that YOU are the problem, just that the relationship is in such a bad place that you being so upset about this is a result of the larger issue.
In terms of the actual issue:
I would have asked the dentist: Is the younger child in pain? Is there any risk for more tooth decay if we wait a month?
If the answers to those questions are yes then I believe health takes precedence over a birthday gift - 18, 21, 50 whatever health comes first.
I also would have asked the dentist for a payment plan up front after explaining the situation. Maybe the whole incident could have been avoided.