I have no doubt that a selenium-based approach has merit, but @AKX's suggestion in the comments to pull the .jsonb file from the Network inspector was the simplest solution.

I hadn't thought to do this, as this is non-standard behaviour for the webtool, from what I have seen so far.

Doing this let me have access to a a bunch of json data wrapped in eval(), i.e eval({...}). Trivial from there.