Now, are either or both of these men (who individually believe in their own mind they are free to come and go at will) actually in possession of FREE WILL?
Ah, the old "gotcha!" example, where you reveal the vital clue at the end and try to trick me into saying something that proves your point that free will is all in the mind, when, in reality, you are asking the wrong question.
The doctor certainly does, unless he is somehow impeded in some way that you are not mentioning. The patient...I don't enough detail. Why is he incarcerated? Why is there a guntower for a mental patient and barb wire? That sounds like a prison to me, not a mental facility. If I presume that we are just talking about a regular mental facility, then I am not in any way arguing that free will for a specific individual (or large group, see China as an example) can't be restricted or suspended, but, for the general population at large, free will exists.