Register the false name with the credit card company as an authorized user of the card. Companies authorize users for "company" cards all the time, so employees can go out and make purchases and sign for them, even though their name isn't on the card. Family members can be added to one user's card for online purchases and such.
Aestu, I've filled out many a "free" account form with a random nonsensical address without problems. When I first started playing WoW, I don't recall having to enter any sort of credit card or billing information. I registered the product key from the WoW CD sleeve, and would enter a game card number every 2 months to keep going. Eventually I did get too lazy to keep going to GameStop every 2 months, and entered a credit card.
Even now, I just signed up for a battle.net account using Johanas Keplerus, with my old spam collector email address,
[email protected].
I entered no personally identifiable information, and it happily created the account. I don't have any other Blizzard product keys to enter (as they're all registered to my main account already), but I'd imagine it wouldn't complain.