When, and to what Extent, does General Relativity Become Relevant

Hi there, guys, I hope this is the right sub, but r/askphysics didn't seem appropriate. There's a TL;DR at the bottom, but first the full version of the question:

I'm a UK high school student with an interest in physics that takes the computer science qualifications at school. Part of this involves a project, and for mine, I have decided to make a Newtonian gravity simulator complete with a visualisation of the evolution of an n-body problem. Part of this task has ended up being that I must include an approximation for the precision of all my results due to both inaccuracies in the numerical integration AND because of the effect of both special and general relativity.

Because I've read around modern physics a little, I know how I intend to take special relativity into account. With each body I will perform a numerical integration of the lorentz factor over the time that the simulation runs, and the special relativistic inaccuracy of that body will be proportional to the result. Obviously I'm not writing a relativistic simulation here, but this should give a rough indication of the extent to which my result can be trusted, since the larger the lorentz factor has averagely been, the more significant the relativistic correction would be.

I'm looking for a similar method to employ to approximate the inaccuracy due to general relativity. I included the special relativistic correction since I intend for the user to be able to (with Newtonian physics) model very fast-moving (relative to one-another) bodies and, since I intend to allow the inclusion of bodies at black-hole mass and volumes, I must indicate how wrong my simulation is here, too.

The best idea I have at the moment (since I know that GR becomes relevant in accelerating reference frames and therefore high-gravity scenarios) is to simply numerically integrate some scaled magnitude of the acceleration of each particle and use this exactly like I intend to use the Lorentz factor, but I have no way of knowing if this is anywhere near the right track. I am also considering integrating some function of the acceleration which (much like the lorentz factor tends to infinity as speed does to light speed) tends to infinity as a body approaches another's schwarzchild radius. My rudimentary research into general relativity shows me nothing analogous to the Lorentz factor, presumably since GR is a field theory unlike SR, and obviously at my level, I cannot manipulate Einstein's field equations in any helpful manner.

Therefore my question is, in short this:

TL;DR: is there anything I can use to estimate how relevant GR is in a similar way to the Lorentz factor in SR?

Thanks!