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?

For any lattice a good choice of a unit cell is the one that is the most symmetric.

Now if every lattice made by a body centered tetragonal unit cell can be made by a face centered cubic unit cell ,which is more symmetric ,then why keep the body centered tetragonal unit cell as a class of Bravais lattice ? • Because not every body centered tetragonal cell is equivalent to a face centered cubic cell. Nov 29 '20 at 15:13

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 "cube" 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.