The simplest way to deal with this situation is to deliver the problem to your card provider/bank. How firm you can be with them is going to depend on how beholding you are to them. Unless you put pressure on your bank they will find it easiest to leave the money in the hands of the merchant facility provider. If upsetting your bank could lead to them retaliating against your banking facility, you will have to be cautious and tactful.
Court action and chasing around trying to unravel who is who from multiple names and identities is a pointless game that won't get you anywhere.
If you need something to quote to your bank you may find this helpful (apologies that my last link didn't appear to work).
http://www.adviceguide.org.uk/wales/debt_w/debt_banking_e/stopping_debit_and_credit_card_payments.htmStopping a card payment
The law says you can withdraw your consent and stop a card payment at any time up to the end of business on the day before the payment is due.
You can withdraw consent by simply telling whoever issued your card (the bank, building society or credit card company) that you don’t want a payment to be made. You can tell the card issuer by phone, email or letter. It's a good idea to also tell whoever the payment is to, that you don't want them to take the payment.
If you read the full article it would appear that your bank were at fault initially and may well have acted contrary to FSA rules in not accepting your original cancellation instruction. Keep in mind that in such a case, if they can't recover the funds, that won't prevent you recouping the money from them.
If it appears that your bank are fobbing you off and don't appear to be dealing with the matter then it is in your interests to raise it as a formal written complaint, which will then enable you take put it to the Financial Ombudsman Service for their adjudication in due course. This procedure will cost you nothing and knowing you intend to do this will give your bank incentive to take you seriously.
As it says in the article, the mere possession of your card details is neither evidence of the existence of a contract, nor authority for the company to use them to obtain cash. If the invoice is not in the name of the company that has actually withdrawn the money from your account then the invoice isn't evidence of any contract either. Bear in mind also that it is highly irregular if your card details are being passed around between separate entities without your express permission.
The benefit of this approach is that the invoice could be your best evidence that there is no contract because you can't have a contract with a fictitious or non-existent entity. If it isn't accurate, informative and precise it helps you.
I am going disregard consumer legislation in the presumption that you are a business of some form.