Why is a face-centred cubic unit cell not regarded as equivalent to a body-centred tetragonal unit cell?
In an FCC lattice  a=b=c while in a body centered tetragonal lattice a=b but c is unconstrained. The first example you gave should be that an end centered tetragonal is equivqlent to a primative tetragonal lattice. There is no end centered cubic lattice as that would violate the minimum cubic symmetry of 3 four fold symmetry axes. however the transition from FCC to a body centered tetragonal lattice would be displacive not transformational.