I decided to try what pendean suggested.

1. I left the Account area of the website and went to Autodesk home page.

2. I picked a product "Inventor" one month subscription $305/month and added to my shopping cart.

3. I clicked on "Check out"

4. The check out page has section to enter payment information.

5. I entered the credit card information and saved the card information.

6. I emptied the shopping card. I DID NOT submit the order for Inventor.

I went back to my Account page, then Billing and Orders, then to Payment Methods. The new credit card shows up.

Feels kind of like driving around the block to go next door, but this disjoint process does work.