There’s a “make default at Apple” button in the card settings that I was able to use to accomplish that without copy/pasting. Weird they didn’t just ask during the setup process.
You change the setting in the Wallet app. Tap your Apple Card, then tap the 3 dots in the top right corner. Press the button labeled “Make Default at Apple”.
I looked there. I really don’t have that button. At this point I’ve manually copied the card info for both my Apple IDs but I don’t think I had that button at any point. Maybe it’s because I have separate IDs for iCloud and iTunes.
I also don’t see anything magical here about how to add it to my Apple ID:
https://support.apple.com/en-us/HT209225