It is very difficult to tell from just the student's point of view. It is entirely possible that the teacher is very engaged with her competition students, and less interested in the recreational students who dance once a week, in which case I would say move him as all students deserve the best training regardless of how many times a week they attend class.
It could also be that although this boy says that he knows the work, in reality he hasn't grasped the finer details of the syllabus and so has become stuck in the grade until he knuckles down and gets the syllabus learnt in depth. There is quite a big jump from IDTA grades 2 to 3 ballet in terms of syllabus content. I have a grade 2 recreational modern student who is 11, and although he thinks he knows the work if I asked him to perform the amalgamations or dance he wouldn't be able to remember them even though he knows most of the set exercises and his rhythm work (clapping and marking) is dubious at best. It's a bit of a 'lead a horse to water but can't make it drink' situation with dance exams- the teacher can deliver the material to the students and meticulously give them corrections, but until they start to A: remember the syllabus accurately (not their own version of it) and B: start to apply those corrections it is difficult. Ultimately the child is showing the Examiner not only what they have learnt from their current grade, but that they are ready to begin studying the more difficult work in the grade above. The grade 2 boy I teach also tends to faff about or talk when I'm giving corrections to other students, rather than listening and applying the correction given to the other student to his own work also, which is another good way for students to improve (not that I'm saying that the lad in the OP mucks around!).
I must say that I also give more in depth corrections to my competition students than my rec students - a rec student (1-5 hrs of dance a week) might be told to 'stand tall' or squeeze their bottom in, but a comp student of the same age (6-15 hours of dance a week) would be told to find their 'neutral pelvis position' and to engage their core and gluteus muscles - if I said that to one of my 10/11 year old rec students I would get a blank look. My comp kids have a much better understanding of anatomy and technique and how to apply corrections, because they are hearing it several times a week not just the once like my rec students.
Obviously if the comp students are doing more hrs of training a week than the rec students then they are going to progress and move up faster. It's also part of a dance teacher's job is to make the dancers take some responsibility for their own dance training and conditioning, and if students want to progress faster then they should work at home as well doing flexibility and strengthening work. There isn't anything wrong with just going along to class once or twice a week and doing it as a hobby without practising at home - but then it's no use complaining that you can't keep up with those that do! I find that this is the singular biggest reason why rec students leave my school - they see the comp kids perform (some of whom choose to dance 5 days a week) but then get fed up that they themselves aren't turning out amazing turns, leaps, and tricks in their routines after their 2 hour a week class. They want to get to that level, but don't want to (or are unable to) commit to the amount of hours of time, training, and sheer dedication that is required to get to that level. There is absolutely nothing wrong with dancing at a recreational level and realising that you are doing fine for the number of times that you train each week, rather than comparing yourself to someone who does far more hours, or worse still resenting them for doing so (again, I'm not saying that this is the case in the OP, just that I am finding as a teacher that so many kids these days want the reward without the hard work that goes before hand). For the first time this year I am going to split my rec and comp kids in their graded classes, and I'm doing it for the benefit of the rec kids. I want them to have more manageable goals to work towards and with other children on a similar level, not to be looking sadly at the technique of other dancers in the class who are doing 5 x as much training each week. I just want my rec students to come to class, have fun with like minded dancers and learn as much as they can about dance in the couple of hours or so that I see them. Without the pressure of trying to keep up with my comp dancers.
OP - If the exams are in the summer would he be happy to do the classwork assessment rather than the graded examination, and is this an option that could be put forward to his teacher? Its a shorter version of the exam and marked a little less harshly. I have done these before with students who have struggled with technique or who aren't quite keeping up with their class group, but likewise I don't want to keep back in the grade with the group below? Just a suggestion, but it would mean he could move up to the next group and get his grade 3 certificate with the rest of the class? If he is really set on the graded exam then he may well have to wait until next term if his teacher feels he isn't ready (or find another school, with no guarantee that he will get the exam done any quicker there either). Has his mum tried having a word with the teacher, just to find out what his weakness is that's holding him back - i.e. knowledge and response of the syllabus / musicality and timing / strength and technique? At least this would give an indication of what area he needs to work on. Good luck though, I have been in this situation several times as a teacher, with a student who is bored but still not ready ( my own daughter is a bit like this with her grade 6 ballet at the moment!) and it's the best feeling in the world when they finally take their exam and you can go into class and teach them some new exercises and dances! Even though dance is about repetition, it's also good to break up the syllabus classes with show work, and free work from time to time.