No credit card required
Browse credit cards from a variety of issuers to see if there's a better card for you.
Although I can't say from personal experience, I can't think of any reason that a balance transfer would not be allowed.
If you think of a BT as just a "payment" to a credit account, the original lender doesn't really care who pays them - he's getting paid, that's it. Chances are that no one is watching anyway - it's all debits and credits within their computers unless there is an "issue" such as non-payment and I suspect minor late payments are still handled by the computers (reminder notices, late fee, etc).
.Although I've never had a secured card, even during my rebuild, I've never had a BT refused or rejected