In theory, your school should be able to provide services to diagnose your child with dysgraphia. It's not an uncommon problem with kids. However, realistically, I know that students have passed through my classroom with undiagnosed dysgraphia. In order to get them the aids or supplements they needed (ie keyboards, pre-printed notes, specially lined paper to help with letter formation), I usually had to hope the student had another disability. It's easier to tack on supplements to an already existing IEP (Individualized Education Plan) rather than convince the school psychologist to spend time on what administration considers a minor issue. Truly, if your school doesn't openly offer to test him, then your best bet is to have a sympathetic 4th grade teacher who really fights hard for your student. (4th grade is the first writing test, so administration might actually care about his writing legibility.)
As far as a private diagnosis goes, I believe an occupational therapist is the best way to go. Perhaps people on this board have had experience with some therapists for children in the area.
Because the student doesn't have a highly visible or 'mainstream' issue that effects testing (beyond 4th grade), you might need to fight the good fight. From my experience as a teacher, administration is resistant to making IEPs for kids unless it definitely would benefit test scores for the school. If you request a 504 meeting with the diagnosis in hand and a plan for helping your student in mind, it would probably help considerably.
Teachers will definitely be on your side if you can help them come up with some solutions that doesn't require a ton of effort on their part. Sometimes they can print their notes ahead of time (not unusual for other disabilities) and have blanks for your student to fill in words so that they must follow along. Other options is having the student type, but depending on the skill level, that might just put your kiddo further behind. Dragon Dictation might be an option for your kid, where they talk and the program writes for them. This works wonders in classes where students write paragraphs or papers. Believe it or not, cursive helps as well! Keeping their pencil on the paper constantly helps kids with dysgraphia form letters easier and makes their handwriting legible.