I've found that Cloudflare isn't great at this. I even found cases where my site was failing to load to googlebot (a "good" bot that they probably have the IPs

I've found that Cloudflare isn't great at this. I even found cases where my site was failing to load to googlebot (a "good" bot that they probably have the IPs for) because they were serving a captcha instead of my CSS. So your best bet is setting a page rule to allow those URLs.