One has to define what physics is and what a "law" is in physics, first:
Physics is the scientific discipline that studies observations and measurements in nature, and looks for mathematical models that will fit data and observations and also , this is important, be predictive of new data and observations. ( Otherwise, if not predictive it is a just a map of the data and observations). Physics uses mathematics as a tool, and the models are called theories.
A law, principle,postulate in a theory of physics is an axiomatic statement, which picks up those solutions of the mathematical model that are predictive of new data. If the predictions are wrong the theory falsified and a new or improved model should be found.
So it is using the axiomatic laws of special relativity and the axiomatic law of the speed of light that allows special relativity to be predictive in all frames. (General relativity has different "axioms" , its principles; at the limit of low masses/energies it reduces to special relativity).
And of course postulates, principles and laws depend on the definition of the terms used .
Inertial frame is defined as :" An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted on by an outside force."
So you must see that "Physical laws are invariant regardless." of frame of reference cannot be correct if the laws depend on the frame of reference?
An accelerating frame makes a body at rest move, after all, so it is not the same as an inertial frame. Extra functions are needed to describe its motion not given by the laws within the frame. The laws are frame dependent by construction .
It is the definition you give to the word "law" that is different from the way the physics theories are using them. A law/axiom that declares : "The laws of physics take the same form in all inertial frames of reference" cannot be extended to non inertial frames if the theory is correct. And special relativity has not been invalidated by any of the experiments that could invalidate it.