Generally your best bet would be to open an Envato support ticket, but it sounds like you’ve already done this.
Unfortunately that error is coming to you from code within Enfold, not directly from our API, and we have no visibility into what’s happening there. For us to really help, we’d need to know the specific error message returned by our API, including the URL and request and response headers from the API call. This is not typically something that a theme exposes to end users, so Enfold’s support staff would be best placed to help you here.
It is strange that it only happens on a single domain. It looks like Enfold uses the legacy Marketplace API keys, which are found in your user’s settings page. Are you using the same key for each domain, or a different one? My best guess is that perhaps on the one that doesn’t work there was an error when entering your key (a typo or extra space or quote present in the field, perhaps?). Can you compare the values on a domain that works and the one that doesn’t?
If we can get the data I mentioned above we can dig deeper into the error, but without it there’s not much I can do. I’d get in touch with Enfold’s support to get some help from them.