Well, after a couple of months investigating, we've come to the conclusion there's absolutely nothing we can do about it.
PayPal tell us: "PayPal does not recommend adding a gift card, prepaid card, virtual card, or one-time use card to your PayPal account. Although some of these cards may be accepted, most do not pass our security checks."
VanillaVisa, who are one of the main issuers of these things, state: "Internet, mail, and phone order purchases may require that we have the name and address of the Prepaid Giftcard owner on file. If you wish to make Internet, mail, or phone order purchases, you will need to go to www.vanillavisa.com and enter your name and address prior to performing an Internet, mail, or phone order transaction." They then go on to say "The card may not be used outside of the U.S. or the District of Columbia including Internet and mail or telephone order merchants outside of the U.S. or the District of Columbia." Since we're a British company, that could be causing problems as well, even though our credit card handling is done through the US.
It seems they just don't work for us, plain and simple, and it's down to some online security verification system somewhere that doesn't like anything other than genuine credit/debit cards. You can try, but don't count on it.