I'm so sorry to hear about how your t search is going!
I too am surprised about the ts reactions. Usually even a very challenging t wouldn't start saying things in the initial session where you both are still feeling each other out, about how you're not accepting responsibility for things that have happened to you. A t that I recently left who wasn't a good match had a challenging stance in general, but at least waited til the second or third session LOL!
So I don't think it was right of her to say that. I also don't believe you need to "present well" to a therapist. This is not a job interview. You're not there to be on your best behavior or impress anyone. Therapists should know that someone coming in is likely to be very upset and even cry. Particularly, if they're aware of and compassionate about BPD then they'd know that you struggle with strong emotions and I would hope they'd be open to the expression of those even in a first session. I agree that sometimes private practice ts can be uncomfortable with very intense emotions. That t that I spoke of earlier who worked at a mental health center was very unsettled when I came in one session and couldn't stop crying. When I'm extremely upset like that I tend to move around a lot, make what probably look like weird hand motions and just look overall like probably not the most together person in the world. Well she was all concerned that I seemed "very agitated" and asked if I wanted to take a break in the hallway. When I told my former t who was a very good match this story she commented that this suggestion was really to make the therapist feel comfortable more than anything else.
So yes I think with some ts it's one thing to have nice neat intelectual discussions about emotions but when faced with them head on it can be quite a different thing.
I like the idea about writing out your thoughts for future ts you see. You could start out a session by saying that you've had trouble communicating with other ts you've been looking at, and then maybe give them the letter you wrote about your issues to read if you feel you can organize your thoughts better in writing. I also recently have been on my own t search, and wrote a reflection type thing for myself about qualities I felt would be good in a t. I sent it to a t I'm considering as part of intake info, and will find out next week what she thinks.
Just don't give up. I think the right t is out there for everyone.