I don't make a big deal out of birthdays. I usually spend them the same way I spend every day. So in my world, you're over-thinking this. But I'm also an over-thinker, so I can sympathize with that.
Myself, I would probably be worrying more that J is agreeing to have me there for JP's sake. Have you asked JP whether this was his idea or hers? Are her other close friends going to be there, or just you three?
Even though you don't know what you want to do for your birthday, you must have a gut reaction about whether or not you want J to be there. If you're more drawn to a one-on-one intimate type gathering, then you'll probably just want JP, and J should understand that. If you're the type to have a big bash with all your friends, then excluding J would be a bad idea.
I can see why you're worried about setting a precedent. That's why I like GG's suggestion of setting the precedent in terms of doing what the birthday girl/boy wants, as opposed to setting the precedent for all three of you. That leaves it wide open for you to decide down the road. It also gives J an elegant out if this isn't really her idea. And if there's any chance you won't want her at your birthday, make sure you make it clear to her that if she wants her birthday to be just her and JP, that you're completely okay with that.
I don't need labels to define me. They're sticky and I hate the glue they leave behind.