As Sarah says the check-up is for you as well as for Hunter so a GP is better equipped to do it. Your GP will be able to do everything that is nessescary for the 6 week check-up and discuss any issues you might want to raise. Its probably good to start building up a relationship with your family Dr too.
I think you should lodge a complaint to the hospital. Its a bit scammy sending in a private peadiatrician to you without warning you that you will be charged.