Hi Zack,
I think your concern here is valid (your render with Screaming Frog or any other client is unlikely to be precisely representative of what Googlebot will see/index). That said, I'm not sure there's much you can do to eliminate this knowledge gap for your QA process.
For instance, while we have seen Googlebot timing out JS rendering around the ~5s mark using the "Fetch & Render as Googlebot" functionality in Search Console (see slide 25 of Max Prin's slide deck here), there's no confirmation this time limit represents Googlebot's behavior in the wild.
Additionally, we know that Googlebot crawls with limited JS support - for instance, when a script uses JS to generate a random number, my colleague Tom Anthony found that Googlebot's random() JS function is deterministic (returns a predictable set) - so it's clear they have modified the headless version of Chrome they use to conserve computational expenses in this way. We can only assume they've taken other steps to save computing costs. This isn't baked-into Screaming Frog or any other crawling tool.
We have seen that with a 5s timeout set in Screaming Frog, the rendered result is pretty close to what "Fetch & Render as Googlebot" functionality demonstrates. And with the ubiquity of JS-driven content on the web today, provided links and content are rendered into the DOM fairly quickly (well ahead of that 5s mark), we've seen Google rendering and indexing JS content fairly reliable.
The ideal would be for your dev team to code these pages to degrade gracefully - so that even with JS support totally disabled, navigation and content elements are still rendered (they should be delivered in the page source, then enhanced with JS, if possible).
Failing that, the best you're likely to achieve here is reasonable confident that Googlebot can crawl, render and index these pages - there'll be some risk when you publish them to production.
Hope this helps somewhat - best of luck!
Thanks,
Mike