My DD7 is bipolar. She was diagnosed shortly after our trip in 2008, and some of the behavior that we experienced on that trip factored into her diagnosis. Our trip last August is the only trip we've had since treatment began, and it made a world of difference.
Early-onset BPD can present different issues than BPD that on-sets in adolescence or adulthood so I'm not sure if my experience will be helpful. DD is rapid cycling and experiences mixed-state episodes. It sounds like the person traveling with you is more likely to experience clear-cut manic and depressive episodes.
The best person to ask these questions is the doctor treating your family member. Each person is different in how their BPD presents and the doctor will have insights that no one else can offer. If a manic episode hits at Disney, call the doctor. A medication adjustment might be needed, and it would be better not to wait until the end of the trip to make that. It's possible for a depressive episode to hit after a trip, but it's hard to say ahead of time if that will occur.
My biggest mistakes on our trip in 2008 was letting DD stay up too late and not getting enough sleep. No matter what age, it's important for someone with BPD to get enough sleep to avoid triggering a mood episode. Try to keep as normal a sleep schedule as possible. DD is a morning person so for our last trip I planned for us to be at rope drop everyday, but in bed by 9pm as well. We missed out on Illuminations, Fantasmic, and the MSEP because of that, but it made our days a lot easier. We were staying at the Poly so I did let DD go down to the beach to watch Wishes one night in her pjs so she could crawl into bed immediately afterward.
DD has a lot of sensory issues that go along with the BPD. And WDW can be massive sensory overload. She is particularly sensitive to loud noises so I bought her noise-reducing ear muffs. Anytime she seemed to be getting aggitated, I pulled those out for her, and they helped her calm down. I also had a stress ball that she could squeeze to help regulate her mood when she was aggitated.
Water can be very soothing for a person with BPD. Take advantage of the pools, particularly at off-peak hours, for soothing water therapy.
Make sure your family member eats some healthy meals. It's okay to splurge on some junk, but good nutrition helps to maintain a stable mood.
ETA: I forgot... Carry water and snacks in the parks. Keeping well-hydrated and not getting too hungry is important for everyone at WDW, but especially important for someone with a mood disorder.