It sounds like that lower processing speed is producing some real issues at school. I'd want to know more about the specialized LD program that you mentioned. Will it nurture his strengths while working on his area of weakness? If it only focuses on weakness, you may find yourself worse off.

Did the psychologist recommend any specific accommodations that the school is refusing to implement? Did the psychologist give you the subtest scores for processing speed? If one subtest is significantly lower, it may give clues about what is slowing him down and how to better give him extra support without having to go to the one-on-one LD program.