I find it interesting that you were able to express emotions in session, when you previously reported, unless I am misremembering, that you were unable to. Originally that was your reason for emailing, because this wasn't happening in session.

I'm not surprised that you want to use email to deal with your emotions. It is not unreasonable to ask your T to respond but he's already said he won't. So that is unreasonable, in my opinion, to not take no for an answer. Learning to sit with emotions is a grown up job, and one thing that might help you is to use part of your session to talk about how you're going to deal with the stirred up pot for the next week.

I don't think the ultimatum is the issue nor is whether you are being bratty or whatever. It seems pretty toned down for what you're describing, the label doesn't seem to match the reality. To me "huge bratty acting out" would be setting his office door on fire or something.

As far as I've experienced, things being stirred up and going to the next session with the intention of processing them is just the way therapy has gone for me. When sessions were particularly intense, my T would offer a "containment" strategy at the end that worked really well for me. So it's possible that leaving a session doesn't have to be leaving you "alone," but your T has made it very clear he's not going to respond to your emails no matter the circumstances. Given that you're making progress, I'd be inclined to stick with it to see how far you can go next session, but you can skip with or without an ultimatum or quit or whatever seems best to you.