The arbitrary constant is related to G, which is Newton's gravitational constant, and to the speed of light. If I've understood correctly, if you don't want pi to appear in the field equations, then it will appear in Newton's equations as part of G.

Well, G is an arbitrary constant but Pi is not. So while you can _formulate_ using a new arbitrary constant, it is HIGHLY improbable you reached tbe point of making that formulation without discovering Pi along the way.