Hello and thank you for your question. I will be very pleased to assist you. I'm a practicing lawyer in England with over 10 years experience.
May I clarify with you that the dealer does not hold the V5C (logbook) document please? They should have the V5C minus the V5C/3 section of the form which is the part the previous owner would have signed to transfer the car to the dealer. If they don't have it how do they explain this?
Have you asked if they carried out a car data check when they acquired the car? If they have have you asked to see it?
From what you say you have paid for the car in full, and from what your bank says, twice? may I ask how much you paid?