Part of fence on neighbour's property.
If a registered land surveyor conducted the survey it is most likely accurate, and getting a second surveyor to do a survey will not reveal anything new. The exception to this is if you truly believe that they made a mistake, in which case you should call your surveyor and have that discussion.
As far as moving the fence if your neighbour asks you to, they are within their rights to ask you to do this as you built an encroachment on their land, albeit a fairly small one. If this happens consider trying to persuade them to let you leave it where it is. If they are adamant then you can choose to move it (in the spirit of keeping the peace) or you can refuse and let them take the next step. Typically that means them hiring a lawyer to take legal action against you (which is expensive and time consuming). You do, however, run the risk of your neighbour taking matters into their own hands and removing that portion of the fence themselves. Although this is not allowed, there's not much you can do once the fence has come down.