O.k.- a few things.
First- you HAVE to talk to your DH about this concept of time spent w/ families. You can NOT live your life keeping a tit for tat tally. Focus on quality time, not quantity. And to your point - your mom makes plans, his doesn't. But regardless - you just can't live life doing this. It's petty and a waste of time.
ANd if it's so important to HIM to have equal time, then HE needs to get off his a$$ and make more plans w/ his family....
Second- you need to have a calm talk about why was this plan was o.k. initially and why did his attitude change after talking to his mom? And I stress CALM. This isn't about accusations, but a "can you talk to me why her anger changed your feelings?". And get him to think about it and talk to you.
Third- once you hammer out #2, he needs to learn to not be so open! His mom didn't need to know your plans.
It doesn't really sound like this is a part of the problem, but I will say this - in general, I don't feel b-days MUST BE celebrated on the actual day. You're close to your sister - but you should be able to celebrate her b-day at another time if necessary. It sucks that her b-day is caught up in the holiday craziness, but most adults don't get to always celebrate their b-day ON the day. I'm just putting this out there so that in the future, this aspect doesn't become a bigger issue. If celebrating her b-day does actually get in the way of seeing his family - that needs to be taken into account.