I have a credit card issued by a US bank, but the billing address is back in the UK.
Sorry, but that card won't work. The credit card has to be both issued by a bank in the country whose iTunes Store you wish to use and that has a billing address in that same country. A split card like yours, issued by a bank in one country but with a billing address in another, can't be used in the iTunes Store in either country.
You'll need a credit card that is both issued by a US bank and has a US billing address, or will need to purchase US iTunes prepaid cards.
Regards.