Parents getting pushback from child's doctor

I am a Prek teacher and I have a student in my class who has a lot of behavior issues, and despite working with the parents (who agree and see the same issues at home), the student has made little progress since he's been in my class this year. He is extremely active-- I know students of this age often are, but no amount of redirection seems to help for more than a minute or two. He has a difficult time attending to small or large group work. Most activities we do are very hands on and open ended (no worksheets or coloring pages), and he still can't attend to those. Even with tasks of his own choosing, he frequently bounces off and initiates something else. I am also genuinely concerned for his safety. He is constantly running, spinning, jumping, etc. He falls out of his chair daily. He invades other students' personal space by touching their hair, talking close to their face, giving gisnt bear hugs, etc. He plays rough as well, finding it fun to crash toys together, often resulting in knocking down structures or even actually broken toys. This rough play has even resulted in him breaking a child's tooth. This rough behavior makes students upset with him and often stop playing with him, but he can never connect his own behavior as a reason why they no longer want to play.

One of his parents mentioned she has a friend who's daughter has ADHD and she sees the similarities. Even though he is only 4, I too strongly suspect that he will end up with an ADHD diagnosis. I dont usually like to push the ADHD idea though, since students are still young and these behaviors can be normal. But for this student, I am genuinely concerned for him. His father took him to the pediatrician, who pushed back and wouldn't give him a referral to a child psychologist, saying he's still young and PreK doesn't count as "real school. " His mother was really upset, and she wants to speak with the pediatrician herself. I offered to write a letter voicing my concerns, but I'm wondering if there is anything else his parents or I can do to advocate for him? Or anything very specific to ask for? This is a kid who is going to be in the principals office daily when he gets to kindergarten, and I just don't want him set up for failure if he can get help. He's a funny, caring, well spoken kid who I think will do really well if he can get these behaviors under control. I will also suggest having him be evaluated through the DOE, but I'm really hoping he can get something like play therapy, which, as far as I know, isn't provided by the DOE, and there's no way he'd complete the evaluation process and be able to get a SEIT before the year is over.