Well, if you provide an ultimatum, you need to be prepared for the consequences. Your T will likely tell you he'll see you in two weeks then. Giving in to your ultimatum would be rewarding bad behaviour.

Is there another way to frame this?

Can you send your T an email, explain that you are aware he doesn't reply to emails outside of session, but you felt this was important enough that it warranted sending. Tell him you are willing to pay him for his time should he be willing to reply, and that you appreciate him taking the time to read it, either way.

Why not try working with him, instead of against him? It's ok to be mad at him - we all get mad at our Ts from time to time, but why not work through that anger with skills instead of pettiness?