The extended illness could easily account for slow weight gain But what a worry for you.
It sounds to me like she is doing very well. I would certainly be looking to get a second opinino - your GP sounds like he has made a bit of a snap judgment, but it is good to be sure of thibgs,
My DS was 4.29 at birth and barely passed 9kg at 12 months, so he dropped a lot on those scales too.