Certainly OP has their own opinion and understanding about it but the way I understood it, OP wasn't exactly using emails to share information but rather found themselves writing angry and out of bound stuff and I believe that is the main reason why the T stopped responding.

To me (and I am obviously not a T but I can still form opinions based on my own experience and what I have been reading, right), this suggests to some kind of borderline dynamics, not BPD but personality features according to psychoanalytic formulation. I think there is a huge difference between object consistency seeking behaviour and hostility and rage stemming from borderline features. While in the former case it might be appropriate to acknowledge the emails, I believe in the second case it would be far more important to work towards directing the hostility and rage into the sessions where it can be worked with, rather than letting it float freely in the space between sessions and facilitate it with email responses.

To me the best way to move forward in this situation is try to get this 2nd session per week going and continue to talk about this email issue in sessions. Changing therapists could be an option too but considering that the rage and hostility is lurking somewhere there, then I don't think it would be helpful to find someone who does not allow these feelings to surface because these feelings quite certainly give some symptoms elsewhere in life.