Is there a way to specifically add an Apple Card as the Apple ID payment method? I am not seeing any options to do so.

Yes, Perhaps this will help: https://support.apple.com/en-us/HT205583

I'm not sure where it is on System Preferences in you MacBook, but on iPhone, if you go into Settings > Wallet & ApplePay > and then click on your Apple Card payment --if you scroll down, there's a link that says Make Default at Apple

I would assume there's an equivalent on the macOS version in Sys Pref.

See Change, add, or remove Apple ID payment methods https://support.apple.com/en-us/HT201266