On the CPT archive page I have custom fields setup as "download" buttons. The one field in question has a URL inserted from a 3rd party site. This use to work, but somewher in the process it's prepending the URL with the actual website domain:
I assume you have labelled the correct URL and incorrect URL the wrong way round.
It looks like the incorrect URLs are being treated as relative URLs rather than absolute URLs so the site URL is being prepended, but we'd likely need to take a look at how you have this set up to understand what's going on.
Let me mark your next reply as private so that we can get log-in credentials from you—you may want to create a temporary admin user for us to use that you can later delete. And be sure to have a current backup of your site.
Can you also confirm which archive we should be looking at, and which custom field is being output.