Hi clairstone,
Welcome to the site

I come at BPD from a slightly different angle my Partner has an undiagnosed BPD ex-wife (uBPDxw). But everyone here often finds themselves in similar situations with our family members.
IMO you go on your trip with your brother just as the two of you planned, don't let your sister hold you guys hostage go and have fun. Your sister can visit with your mom just as she planned.
You are not a bad person for going, I can hear that you are feeling obligated. It's a tough place, that space between what you want to do (go camping with your brother) and what you feel you should do (include your sister). I have had some real meltdowns in my past when I have found myself squeezed between those two and what I finally decided for me that I will choose what I want to do not what I'm obligated to do. There can be some guilt by doing that but over all I find I am happier. Besides if I do something out of a sense of obligation I will become resentful.
You are not responsible for your sisters emotions she is, I know it's uncomfortable for you that she is uncomfortable but she needs to deal with her own feelings and you should go camping!
I want to share some of the information from the site about some things I'm hearing in your post. The first is FOG (Fear, Obligation, Guilt) or emotional blackmail. I am hearing all three in your post...Fear that she will be mad at you, Obligated to include her because she's your sister, and Guilt because you really just want to go with your brother. Usually if you are feeling pressure from your sister it's likely FOG...
Link to more on FOG...
https://bpdfamily.com/message_board/index.php?topic=82926.0The other thing I heard in your post was JADE (Justify, Argue, Defend, Explain). We all do it particularly when we are being falsely accused or attacked. The goal with someone with BPD is to keep the JADE at a minimum. When we JADE we just provide more ammunition to up the drama and off we go often ending up in a circular argument that goes no where.
More on JADE...
https://bpdfamily.com/message_board/index.php?topic=139972.0I'm glad you decided to jump in and join us. Again Welcome.
Panda39