How to deal with student with low self-awareness about health condition?
I should not diagnose him

That's correct.  Even if you knew how to, he is not your patient.

I would like him to see a professional

Also correct.  You should tell this student they may benefit from an appointment with the disability office at your university. You can also ask that office for advice.
You should tell your supervisor about the problems with the student's performance.

Should I continue helping him?

If you have tried to help a student several times and made no progress, it is acceptable to give up.  This is particularly justified if the student refuses to follow your advice.  Depending on circumstances, you may also be able to give the student an easier research project.