Hi,
This sound more like a CDN problem, but I am not sure yet. Once a CDN is active, it should cache the files to it’s remote location and serve them instead of the local files. The search has no control over that whatsoever, and all the assets are registered and loaded according the wordpress guidelines, so I don’t know what the cause might be.
Error 502 might indicate, that the CDN tried to load the files from it’s own gateway, but they were not yet cached/registered, or something similar. At the moment I am seeing the search assets loaded from the ‘file.’ subdomain on your site, and no errors on the console, so I am not sure if it’s already resolved.