Y beat me to it and gave you good advice. You don't ignore you just don't add and initiate. Reconciliation if possible is likely years down the road and she is not going to think anything about you being brief in your small talk.
Why let her cake eat? How will you feel if you are out to eat and she gets a text from OM? She lost the right to celebrate your birthday with you.
It is better that you start to get used to doing things with your kids alone.
