Choc, stop saying you shoulda/coulda done something else. Everything you have done has been exactly what you needed to do at exactly the right time. This whole thing does NOT hinge on the timing of one email or one comment. Give yourself a break.
Choc, you CAN miss your dad's birthday. It IS an option. What if you were the dad and one of your married children needed to miss your birthday because he was needed at home? Surely your dad will understand? Can you go at another time, maybe a week or month later when ALL of you can go? I'm not saying you should skip the birthday, just saying you should consider it, make it a real option. Don't automatically slam the door on that option, for it IS a real option if you feel you are needed at home. OTOH, maybe going is the best thing for YOU. You have to do what is best for YOU. You can't be everywhere at once and be everything to everybody. Just think it through.
This is what to focus on: "For a time, at least a short time, you show her what she is potentially giving up (the good Choc), and her intact family."