They really have made it complicated.
Despite being a fluent reader of Danish I had completely missed that it was at all possible to get the personal card without a CPR number. Do you have a link to more info on that?
"Fun" fact: The underlying technical specification of the card
is identical to that of the Swedish regional transit authorities (Swedish). However, since they haven't managed to ensure interoperability within the same country (except for a few counties in south Sweden: Kronoberg, Halland, Blekinge, Skåne), I'm not holding my breath for this to work cross-Sound any time soon.