So most of the hard work was in this repo: https://github.com/ruflas/crunchyexporter-cli. I'm just using it to export my watch history so I can keep track of things I start but forget to finish.

From there, I can serve it up on my own web app that then connects to my Apple TVs via https://pyatv.dev. So from my phone, I can browse my Crunchyroll history, hit play and it starts playing on my Apple TV, all without hitting the Crunchyroll apps or site.