Should non-relativistic Navier Stokes Equations be modified so that they become pseudo-Lorentz invariant?
There is an extended literature on relativistic dissipative hydrodynamics, modifying the Navier-Stokes equations to accommodate relativity. There are a number of inequivalent ways of doing this, only some of them leading to a causal theory. A paper by Van and Viro https://arxiv.org/abs/1109.0985 (one of many) gives lots of recent references. See also the book ''Rational extended thermodynamics'' by Müller and Ruggeri.