There’s like a billion side channels to determine how big the screen is unless you just want to entirely break basic css. Which is a pretty unreasonable way to address this problem.
Surprisingly, most sites are perfectly usable with CSS disabled. They end up looking a bit like "motherfucking website"[1], or what you see in a text-based web browser.
Wouldn't loading all external links right away (think background-image) solve this? How does the site exfiltrate the gathered information without javascript or tracking pixels?
Edit: Having a bunch of html buttons/links, showing a different to agents based on their resolution and waiting to see which ones they follow would break this, unless everyone crawls a lot of stuff they don't need. Pretending to be one of a few common sizes is probably a better solution.