Baby Lisa's father claims negligence and fraud - but who's fraud?

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MSNBC reports that the parents of missing baby Lisa Irwin are accusing police of neglecting a "fraudulent debit card charge" which they claim has gone uninvestigated.

"In early November, just over a month after Lisa vanished, Irwin said his debit card was stolen. The card was suspended for a fraudulent charge for $69.04 while two other charges were attempted. He said the charge was made to a British website that advertises a service to legally change your name or your child’s name online."

A quick google search and I found this: https://www.namechangeco.com/products/deedpoll/

A UK company which sells a document kit which allows you to create a document to accompany a birth certificate and to provide a complete change of name for a child.

It costs £33.00 which I believe is roughly equal to the $69 quoted.

Why would a baby thief do this?

1) You need an actual birth certificate to modify to use this kit. If you had an actual birth certificate why would you need to change it? I don't recall hearing them say they stole baby Lisa AND her birth certificate.. AND Jeremy's credit card!! oh my.

2) Purchased from England? One month after the kidnapping? A bizarre related, but illogical purchase, on the card of a father of a "stolen" baby?

Pop Quiz: Which scenario is more likely?

A baby thief steals a baby, and the baby's birth certificate, and the baby's father's credit card.. smuggles the baby to England and then only one month later uses father's card to purchase a cheap document kit to create a new false identity on top of stolen birth certificate for stolen baby ... to create a new fake identity which stolen baby won't need for five or six years until she is entering school.

OR

Guilty parent, feeling unwanted attention upon him and feeling the pressure of disbelief of his story, uses his own credit card to make a bogus purchase of a useless document kit in order to make it appear that someone in England was trying to change the name of his baby?