So it appears that the answer is that the bulges are a forcing function, not a displacement.

Am I the only one skeptical that Newton would confuse a force with a displacement?

The term “newtonian solution” just implies that you’re using gravitational and kinematic approaches to the model. I think Newton probably would have done much better had he made an earnest study with the resources he had available. People in shipping towns knew when tides and currents would be favorable and ships would try to leave at those times.