So far as I know that classical theory has not advanced since Trump and Schieve published their monograph and, as they admit in the monograph, the current theory only can solve some weak-field defects of general relativity.

There is some interest in the application to many-body quantum bound problems, where quantum field theory also fails, and some recent attempts to apply the new relativistic theory to an extension of string/brane theory (which inherits the defects of both quantum field theory and general relativity): The Landscape of Theoretical Physics, A Global View: From Point Particles to the Brane World and Beyond in Search of a Unifying Principle. But as said before the whole theory is still open to many technical objections.

Another attempt to solve the defects of field theory is by Chubykalo and Smirnov-Rueda

See specially the Physical Review E paper correcting the defects of Maxwell theory of electrodynamics and their experimental application in Journal of applied physics.

Chubykalo and Smirnov-Rueda approach has been recently extended to many-body gravitation, with the bonus that the new potentials solve the dark matter problem (dark matter is fictitious). See Modified Newtonian Dynamics and Dark Matter from a generalized gravitational theory