Your main responsibility is your immediate family: your husband and child. The dedication is a major milestone in your child's life and should be celebrated. However, this does not mean that your sister-in-law needs to celebrate with you. Likewise, her main responsibility is to her child and at this point in time the most important thing in her life is her child's birthday.
It is understandable that you are both hurt by the situation, however, arguing or responding to her irrational behavior is only going to fuel the fire. Don't try to be right, arguing only gives her more of the drama she wants, just be polite.Thank her for the invitation to the party, politely decline, send a gift, and put the situation from your mind.
Enjoy your child's special day and don't give the sister a second thought. Just because she is family does not mean she has to be your friend; if the relationship is worth having, then she will come around again.
Best of luck to you :)