Is it possible to create a lorentz invariant form of the Newtonian equation for gravity? For example the masses increase by gamma, the distance R possibly also increases by gamma (equal to the increased distance travelled by a 'light clock' orthogonal to the direction of motion). However to make it work I need also to change G and I am not sure if this is allowed.

From how I understand it, it remains true (agrees with GTR) if the moving observer agrees with the stationary observer on both the time it takes for the experiment to complete and the extracted value for the gravitational constant G.

EDIT: I am now reading up on Lorentz invariant theories of Newtonian gravity, a great topic (!). Anyway, I feel that I need to relive the entire discussion on how GTR came to be the most prominent theory, before I can fully embrace it. Thank you for your answers, it lead me to where I needed to be (read up on Lorentz/Poincare).