I actually got this working.
(First, make sure both accounts have two-factor authentication enabled. Otherwise, nothing from your wallet can be added as a payment method.)
The first thing I attempted to do was add the Apple Card as the payment method for my iTunes account, which worked. At that point, I went to iCloud payments and noticed that the "Found in Wallet" section did not have Apple Card as an option, which is the problem you are having. So, I went back to my iTunes payments and removed Apple Card as my payment method. Then I went to my iCloud payments and saw that Apple Card was now available, and I set it as the payment method for iCloud. Then I went back to iTunes once more and saw that Apple Card was still available here, and added it as well. Now I have Apple Card for both.
tldr: Add Apple Card to iCloud first, then add Apple Card to iTunes second.