Aha! There's a way to cheat!

The old Twitter embed API lets you manually specify colour scheme - and is easy to screenshot.

For example: https://platform.twitter.com/embed/Tweet.html?dnt=true&embedId=twitter-widget-0&frame=false&hideCard=false&hideThread=true&id=1420704988825800704&lang=en&theme=light&width=550px

Hopefully I can replace all the Twitter embeds on my website with images (and, yes, I will add alt text).

Thanks @samir for giving me the inspiration.