So what you are describing is actually enmeshment.
You live in the same place, have the same friends, go to the same school. So it is basically impossible for you to have any kind of separate life or identity.
This is really common with twins. Some have no idea how to really live or survive without the other, they are like two of the same person - but are of course not the same person at all. So it creates conflict.
In your place given your predicament. I would go to the weekend away absolutely.
You can prepare the ground now for this by organising coffee individually with each friend alone and explain you feel out of the loop with your new baby and would like to feel more part of things again. Be honest and congruent about how you are feeling. Build up your friendships individually again and quietly, no need for your sister to be involved.
In your place, I would organise a special birthday treat for yourself and your own little family besides the weekend away. That way if it is a bit rubbish, or all about your sister then you won't care as much, and it will be easier to let it go. Plan something just for you op.
Please don't ever expect people to take sides, ever. They will never do it. It would be a cruel thing if they did. They care about you both. They won't want to be involved with anything that could cause a rift.
Start developing interests and hobbies outside of your current group, so that you can have more choices if things were to change. Make new friends, be open to new experiences. Don't live your life in the twin bubble. Be separate sometimes, but not in a drama way that might cause hurt, but by simply expanding your horizons.
I am the dd of enmeshed twins, please don't forget your kids in all of this, it should not always be about your sister (my stuff I know) They want some part of you as well. Make time for them.