It turns out GoDaddy only issues the certificate to the domain attached to the hosting (which is somewhat of a stupid association, really, because it shouldn’t matter what domain you use as long as you’re accessing your content via a correct route).
If hosting with GoDaddy was the first mistake, this assumption was the second. The whole point of an SSL certificate is to say that the server you are talking to is a server that is authorized to speak on behalf of domain xyz and, as such, they are generally tied to one or more domains.