Gravito-magnetism and gravitational waves outside of General Relativity

An important implication of (linearised) Einstein equations is that you can write a gravito-magnetic field in addition to the gravito-electric field (the classic gravitational field). And from there you can also write the analog of Maxwell's equations: ok, they're not quite the same because of the sign difference and because of the tensorial nature of gravity, but they're very similar.

I'm wondering if General Relativity (or any surrogate "advanced" gravity theory) is really necessary to describe the gravito-magnetism and the gravitational waves phenomena. To me the fact that there should have been a gravito-magnetic field was clear since when I studied Special Relativity and I learned how the magnetic field can originate from the Lorentz boost of an electric field.

So basically my question is: can we describe gravito-magnetism and gravitational waves without using General Relativity, perhaps just by considering an accelerated frame of reference? (I'm also thinking about the Equivalence Principle here, to see the action of gravity reproduced by accelerated frames)

Gravitoelectromagnetism (GEM) can be derived as a hypothetical exact theory analogous to classical electromagnetism. The idea is to take the Maxwell Lagrangian density and flip the sign of the $$F^{\mu \nu} F_{\mu \nu}$$ term; this results in an attractive force between like "charges" (masses). You can do this without knowing anything about general relativity. Unfortunately, as explained in the MTW Gravitation textbook (exercise 7.2 on page 179), this "vector theory of gravity" predicts that gravitational waves carry negative energy, which ultimately forces us to admit that it can't be the correct fundamental description of the gravitational field.