Just ask the customer to send you their finger

It's likely not recoverable. I haven't worked on these, but I'd be pretty pissed if I bought one, thinking my data was secure, and it could be bypassed by taking the drive out of the enclosure. I see one of four scenarios:
1) The data is encrypted by a single key, determined at the factory.
2) The data is encrypted with a random key, stored on the drive itself
3) The data is encrypted with a random key, stored in flash,
4) The data is encrypted with a key determined by the fingerprint itself
If it's #1 or 2, there might be a utility somewhere that will decrypt the drive. If I was LaCie, I'd keep such a utility under lock and key.
If it's #3, you'll need a method of decryption, and a way to read the flash.
If it's #4, there will be no decryption without the original finger.
This all assumes they used strong encryption. Some research might produce answers to these questions.