"Nice man" is evidently concerned to protect his no-claims bonus, and I suppose that if the accident were to go on the record your motor insurance might also be affected. However, isn't there a legal requirement to report accidents to the police?
Do bear in mind that if any damage (to the car or to husband) were to come to light at a later date, then not involving insurance would mean that you would bear the entire cost for yourselves -- and physiotherapy is not exactly cheap!
To agree with the others, however, giving your bank details is safe.
Police no, insurance yes