I did exactly the same thing for my own records (don't have kids). The GP didn't want all the records but he did want my immunization records. I think I had to pay my US doctor to get the records since I was picking them up and they weren't being sent directly to another doctor but it wasn't much and was definitely worth it as a couple specialists have requested my previous medical records. So, I would suggest getting your own records in addition to your child's records.