Is it permissible, Islamically, for a legal notary to take an amount exceeding his agreed-upon fees if the client thinks the agreed-upon amount is higher and pays it willingly and thankfully, knowing that the notary's fee is not fixed, there is no coercion from the notary, and what the client paid might be less than the customary fee?

If the two parties agree upon a specific amount before the work begins, there is no objection to taking the agreed-upon wage, especially if it is less than the customary wage. Perfect sincerity dictates informing the other party of the wage you are content with, even if it is less than what others might request, following the example of Jarir ibn Abdullah, who pledged allegiance to the Prophet, peace and blessings be upon him, to be sincere to every Muslim.