What would a specialist be called whose job is to convince people?

Solution 1:

In the situation you are describing, the person is called a Closer.

From Dictionary.com a closer is:

a person or thing that closes

Which isn't terribly helpful at all, until we look for the correct disambiguation of Close:

to arrange the final details of; to conclude negotiations about

and

to complete or settle (a contract or transaction); consummate:

These definitions are probably the best fit for the term "Close" in the context provided.

Could also be a lobbyist, or if talking about a group of lobbyists just a lobby. Also from Dictionary.com:

a person who tries to influence legislation on behalf of a special interest; a member of a lobby

Solution 2:

Especially in business, this person is often called a closer.

Closer: A person who is skilled at bringing a business transaction to a satisfactory conclusion.

Example: Harvey Specter in the TV series Suits is called the best closer in New York because of his ability to come to agreements with his clients.