An example shows that Maxwell Equations do not need Galilean invariance, therefore Lorentz Invariance is not needed for Maxwell Equations. The example: In a train moving at a constant speed relative the earth, there are electrical charges static relative to the train. In the train, an observer can only observe the electrical field, no magnetic field can be observed. Out of the train, on the nearby ground, an observer can observe the electrical field and magnetic field produced by the electrical charges in the train. This example showed that in different related inertia systems, the same electric and magnetic experiments can show different results. These different results are not mere space dimension difference caused by the relative movements of the inertial frames. These difference is physical electromagnetic property difference. Maxwell Equations do not need to have Galilean invariance which is a space dimension invariance. Maxwell's equations need to be different in different related inertial frames in order to correctly describe a same electromagnetic experiment observed in the different inertial frames. Therefore Lorentz Invariance is not needed for Maxwell Equations. For the same reason the Relativity theory is not needed. Because Lorentz transform tries to use space dimension change to mask the physical electromagnetic property difference in Maxwell Equations.
There's a misunderstanding here. Maxwell equations are not invariant under galilean transformations. Physics needs to be the same in any inertial frame, thus we need to define a new set of transformations. This is why we introduce the Lorentz transformation : it leaves Maxwell's equations the same in any inertial frame. Or, if you want, we need all the terms of the equations we write to be Lorentz covariant.