It depends on the context of the situation.
Negotiator is probably the closest general word in common usage for what you are describing. That, however, carries implications that the final result will be some sort of deal and that there may be some give and take. That would not be the ideal word without those connotations.
You have essentially provided the job description for a Lawyer, but that word implies there will be some interaction with legal institutions or other lawyers. It may also imply that the person being referenced has a license, depending on the jurisdiction being discussed.
A Lobbyist will perform those functions, but that term is almost exclusively used in connection with politicians or similar bureaucracies.
A Consultant may play a similar role. Normally, a consultant will be brought in to help determine the best course of action rather than to persuade a person that a previously chosen course was correct. However, the end result may be the same.
Another answer suggested Influencer. I agree with this, but influencer is a relatively informal term and also one that carries connotations of influencing large groups of people. I have never yet heard of an influencer being brought in to target a specific individual.
Closer was also suggested. Again, I agree but this is also a fairly informal term that carries some significant connotations. In the contexts I am familiar with, Closers are usually involved in sealing a deal and the word would sound odd used in a context that was not expected to end with a contract or transaction.
Rhetor or (with credit to Ooker) rhetorician also come close in their literal meanings. The words refer to someone who is an expert in rhetoric which is essentially the art of persuasion. However, those words are not in common usage and many native English speakers may be unfamiliar with them. They now mostly comes up in discussing historical teachers of rhetoric which was common in some time periods of Ancient Greece and Ancient Rome.
(With Credit to Brian Drummond) Evangelist has a similar meaning. Originally, it was used almost exclusively in religious contexts, but the meaning has been broadening. It comes up often in technology now. However, like influencer and some of the others, evangelism is more often associated with trying to sway large numbers of people rather than trying to persuade a specific individual.
Advocate refers to someone who presses a specific position and comes very close to your description. However, advocate is sometimes used a synonym for lawyer (along with many others including counselor, attorney, barrister, etc.) When it is not being used in the sense of being a synonym for attorney it does not necessarily carry an implication of skill or training the way that negotiator, lobbyist, and consultant do.