He is absolutely in the gifted range, and he absolutely has motor or processing issues, and, based on the other things you are saying, possibly dyslexia or dysgraphia as well, and it is totally inappropriate for the teacher to say that he just needs to try harder. The GAI is almost certainly a better indicator of his intellectual functioning than the FSIQ because of the very low processing speed score. You should request an IEP meeting if you haven't already. Get the psychologist's recommended accommodations for him in writing on a legally enforceable document (the IEP) now, and ask to get Occupational Therapy started as soon as you can. You might also want to ask to get his visual processing and visual-motor coordination evaluated, as this can cause reading problems as well as writing problems. A dyslexia evaluation would seem to be in order, too, if it hasn't been done. If you request these evaluations in writing, the district has to comply within 45 days.
You might also want to to check out
http://www.wrightslaw.comThe sections on evaluation and testing, eligibility, writing effective IEPs, and Twice-exceptional students (gifted with learning disabilities) are probably the most important places for you to start with that site right now.
His IQ scores show that he is functioning intellectually at a level higher than all but one or two percent of his same-age peers, but his processing speed subtest score (which can also be an indicator of fine motor planning and visual-motor integration) is worse than 96% of his same age peers. This means he will have a really hard time showing what he knows and how well he thinks on his written work. If his teacher values speed and neatness over creativity (and it sounds as if she might), he is likely having a very hard time in this class, and may feel that he is not smart because he can't write well and the teacher doesn't like his ideas. An environment like this can lead children to avoid work or stop trying, which reinforces the teacher's view of the child as not very smart. Hopefully, the school psychologist at the IEP meeting will be able to set the teacher straight.
Best of luck.