How do you convince someone to see a neurologist?

If he is a competent adult, you can only tell him that you'd rather he get to see the doctor of his choice and not the one the court orders if he declines further.

What does he fear? My mom thought they were going to say she was "crazy" and she would lose all her autonomy.

You say he has been to his regular doctor and that no testing was done. Is this because he refused any testing?
Does your brother have a history of mental illness?
How old is your brother?
What symptoms are you seeing?
CXMoody has a good suggestion about asking him to do some simple tests.
Is there any substance abuse of any kind?
How long have these changes been happening and are they sudden or having they been coming on slowly?
Without a diagnosis and without your having guardianship there is little you can do. If you feel your brother is a danger to self or others do call Adult Protective Services and ask them to do a wellness check on your brother. They do have the power to intervene if they feel they must.

Would he be willing to do one of those paper/pencil screening tests?

You can find them and administration directions on line.

MAYBE he would be willing to take one just to “prove” to you that there’s nothing wrong?

Is he afraid of losing his driver’s license, or other fears?

Has he been seen by his primary cafe doctor?