Is there a word for a professional who has a beautiful and neat handwriting and whose work consists of using that attribute?

Solution 1:

A calligrapher. There is the much lesser used word, "calligraphist."

Solution 2:

Calligrapher

1: a professional copyist or engrosser

2: one who practices the art of calligraphy

Merriam-Webster

Solution 3:

Calligrapher, as suggested by other answers, is most suitable for the majority of uses. A couple other words may apply, depending on the context:

• An engrosser is someone who copies out an official document in fair copy, such as was done for the United States Declaration of Independence or the Constitution of India.

• A penman is essentially a synonym for a calligrapher, but may imply more of a skilled hobbyist than a professional. The term master penman has currency in calligraphic societies. The term gives rise to penmanship, a word describing (usually good) handwriting, especially everyday handwriting.