Hi, I am a UK doctor.
To my knowledge there are no UK child psychologists on this site. We are all doctors.
However as a UK GP I have to tell you that there are no medical or social grounds here for admitting your son to a bilingual school.
Unfortunately you have even less of a case than if you were Spanish and coming to England.
As both parents are British, and I assume your son is registered as a British citizen, the fact he was born in Spain would not be considered a reason for him to have a medical or social reason to attend a bilingual school.
I realise this is not the answer you want to hear but child psychologists see children with learning difficulties, behaviour problems and the such like.
The circumstances you describe would not be the remit of a child psychologist.
The best you can do is to make your case with the individual schools.
I joined JustAnswer in June 2013 and am a UK doctor. I have been qualified for 23 years and have widespread experience in GP and cancer care.