Why does the barium ion in barium titanate move in its tetragonal crystal structure?
This is known to be because the $\ce{Ba^2+}$ ion sits off center in the middle of the tetragonal crystal structure


First of all this is not correct. It is the $\ce{Ti^4+}$ ion that sits in a tetragonal off center configuration (black atom in perovskite structure below). Also this hold for any material with perovskite structure with the $\ce{ABC3}$ general formula where $\ce B$ is your center site, $\ce A$ is the corner sites and $\ce C$ is your face site counter ions $(\ce{BaTiO3, LaAlO3, NaMgI3, ...})$.




  Additionally, does the $\require{enclose}\enclose{horizontalstrike}{\ce{Ba^2+}} \ \ce{Ti^4+}$ ion naturally sit off-center in this configuration, or does it only behave that way when introduced to a charge?


It does sit off center all of the time in any of 6 directions on a $(001)$ plane but the many centers are randomly oriented which produces no net dipole moment. When charge is introduced, the $\ce{Ti^4+}$ are attracted toward the negative plate and will shift toward that plate creating a net dipole moment.


  How much freedom of movement does this ion have in a solid crystalline structure?


Very little, the $\ce{Ti^4+}$ is located between the plane of 4 oxygens (red atoms) and an oxygen atom perpendicular to the 4-atom plane. It can move between the 6 possibilities of this shift $(\pm x, \pm y, \pm z)$ but not between cells.


  How does this $\require{enclose}\enclose{horizontalstrike}{\ce{Ba^2+}} \ \ce{Ti^4+}$ ion, or even the crystal structure itself, move in order to accompany a larger amount of charge?


When the electric field is applied, the titanium ion receives enough force to squeeze between oxygen atoms to a new lower energy position (e.g. $-x$ to $+y$).