The first step of doing a calculation is finding a metric. The most realistic choice is NOT going to be Kerr or Schwarzschild, because the Earth is not an idealized black hole. Since the OP is still asking about which metric to use, I don't think they are prepared to do a calculation as of yet, and that if they think about the steps they need to do a calculation, the first one is to find a metric.
I'm pretty sure the OP has at least seen Neil Ashby's rather famous paper on relativity in the GPS, for instance
https://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf. But really, there are a lot of possible choices, and it's the OP's job to make one. My impression is that they are struggling to conceptually answer the question of what metric to use, I can and have made some suggestions that might be interesting.
So the first thing I would point out is that this paper does (IIRC) give some suggestions as to a metric.
The ones I would suggest are Ashby's paper (not my favorite, but the OP uses a lot of language from it, so I think it's a paper that they've read that may be important to their thought process), Misner's "Precis of General Relativity" which is basically a response to Ashby's paper (I prefer Misner's approach over Ashby's, it's at least worth reading). I'm also partial to MTW's treatment of PPN theory in "Gravitation".
Honoroable mention should be given to the IAU's approach, as a highly accurate one that's of obvious official interest, but it is probably not the best place to start. This has several revisions, so one would have to pick one.
As an overview: Basically rather than using an exact analytical solution that doesn't fit the actual Earth , such as Schwarzschild or Kerr, I would use some model based on the linearized gravity.
After one has picked one (or more) metrics of interest that one believes model the metric of space-time around the Earth, then one is in a position to do one or more calculations and compare the results and look at the significance of various effects.
I'm not quite sure why the OP is so focussed on coordinates, it's rather like they know that methods independent of a particular coordinate chart exist but they've forgotten this? Anyway, this is all a bit of a reahsh, unsuprisingly.
I'd also suggest looking at what we know about the Earth, for instance
https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html, and suggest that some basic parameters of interest are GM, and J2. A brief theoretical diversion into into "potential theory" (see Goldstein's classical mechanics, the chapter on the Earth-moon system and the figure of the Earth) might help explain why J2, a ratio of moments of particular interest.
Anyway, I'm not holding my breath, but maybe we can move past "Should I use Schwarzschild or Kerr" :).