IT'S WORKING! Partially.
I started walking the problem backwards, reasoning out what variables I was going to need to move the character to another space. When I got to the point where I was looking at what the move action would be I realized that iTween's Move To action would accept a GameObject. That meant that one of the three variables I thought I would need, a Vector, wasn't actually needed.
I started by building the Game Controller FSM. I collect the Tile Game Objects to set up the TileArray, associate a slot on the Array (#2, which holds Tile 3) with a Game Object (CurrentTileObject), then associate that Game Object with an Int variable (CurrentTileInt), and finally establish NextTileInt as just being 1. I then used iTween to snap the player to the variable CurrentTileObject. He popped right over to it. So far so good.
I then spent the next three hours building and rebuilding the actual Player movement script. On a key press the script does an Array Get Next and starts with CurrentTileInt and ends on NextTileInt. It finishes by associating NextTileInt with NextTileObject. The next state adds the CurrentTileInt to NextTileInt, sets CurrentTileInt to itself immidiatly after, then do an Array Get to associate CurrentTileInt with CurrentTileObject. The last state is another iTween, which now moves the player from wherever they are to the new position of CurrentTileObject.
I had the FSM done about two hours ago, but when the player moved, he would get to the correct tile, and then snap back to his original position. I reworked the whole thing a dozen times or more, trying a bunch of configurations, and getting the same result every time. I was pulling my hair out until a friend who had given me some early help asked if the script he gave me to run the player early on was still active. Sure enough, it was, and once I unchecked that the player moved and stayed in place.
Of course the next thing to do was to reverse the process. I figured I could use a -1 for the NextTileInt to move back as space on the Array. Wrong. I get locked in the first Array Get Next state.
I'm not entirely sure how I'm going to fix it, but this is a huge step forward!
Thanks again Mdotstrange!