If a good choice of a unit cell should be the one of most symmetry ,then why keep body centered tetragonal if face centered cubic exists?
Body-centered tetragonal is face-centered cubic only if $c/a=\sqrt2$.  If you try your transformation with a $c/a$ value greater/less than $\sqrt2$, your &quot;cube&quot; will have lateral edges that are longer/shorter (resp.) than the basal edge and so really remains tetragonal.
You can also read this the other way.  If you are presented with a body-centered tetragonal lattice and $c/a$ seems to be close to $\sqrt2$, maybe it's really fcc.