Is there a usage note for choosing between these two phrases? Nothing in Garner. And I've seen it both ways.
To that end, we propose the following compromise.
To this end, we propose the following compromise.
English Language & Usage Stack Exchange is a question and answer site for linguists, etymologists, and serious English language enthusiasts. It only takes a minute to sign up.Sign up to join this community
In this sentence, end does not mean side, it means goal. To this end means In order to achieve this goal.
Whether you say in order to achieve this goal or in order to achieve that goal makes very little difference, if any.
If your sentence was to win, he has to run to this / that end of the trail, there is a difference, with this, he has to start from the far side and come towards the speaker, whereas with that he runs away from the speaker to the far end of the trail.
'To that end' sounds conversational. 'To the other end' may be apt.