Yy. Tbh it is reasonably common for yr r and yr 1 kids to struggle with the sharing the teacher's attention stuff, and often has v little to do with being gifted - kids get a lot of 1-1 time with mums/ dads and in the smaller ratio settings at nurseries, and so the transition to being one of 30 can be problematic! It's also reasonably common for nt kids to do what they are told at school and then let rip in the safety of home, where mum is a known quantity! 
I always find it interesting that they would rather go for pathologising the child, rather than looking for the often simpler solution of challenging them, to see if frustration with a boring and too easy curriculum is the issue. I'm assuming there are other flags they are considering which point more towards sn rather than gifted being the main issue, in this case.
Where the gifted issue is a genuine possibility, I see no particular virtue in concentrating on the sn route, though, tbh. At least not exclusively. The sends could quite easily have put him on the waiting list for the Ed psych that covers the school, for assessment. Interesting that they are pushing the ball back into your court for sn follow up. (gifted is the very lowest priority on school's radar for Ed psych. On the early years they are playing catch up with un-dx sn).
The route to dx is very long and there is no quick answer to everything. (plus, if he is as high functioning as he seems, then there will be little or no support for him that comes with the dx. So you will have a label, and an answer, but the practicalities of living with it will be very much down to you... And to what school are willing to put into place.) a lot of parents of high functioning kids look to read up on possibilities and start to work out coping strategies etc prior to dx.
For example, as the second has suggested possible aspergers, what difficulties does she see in school that he needs support with? Has she and the class teacher put an iep in place with realistic targets for him to be working to? He should already be listed on the school action register, with the referral to developmental pediatrician putting him on the school action plus register. So, he is an sn child. What targets does he have on his iep? At this point, they may choose to concentrate on the social side - and look to him learning and following the 'rules' of the classroom, if he is academically able.
And later, of he does turn out to have a dual dx, he should have iep running for both his gifted and sn targets.
Why are you waiting for school to write to the gp, out of interest? Far quicker for you to make an appointment and ask for a referral yourself? the letter will follow and can be added to the paper trail.