I'm talking about the full calculation, including perturbations from other planets. I've seen the general relativistic correction done a half dozen times before, but I can't say that I've seen the whole thing done. Maybe I'm just bad at Google, but every search I make just comes up with the GR correction. Does anyone know where the full calculation is done?
After a bit of research, the key term here is "the secular dynamics of Mercury". With that, you can easily find course notes that cover the whole calculation:
It's frowned upon to give a link-only answer, but this is a big reference so I think it's appropriate.
The thing that would have been done in Einstein's time would have been to take some sort of first order correction to Newton's equations, and then a whole lot of perturbation theory.
Note that the geodesic equation for $r$ only differs from the newtonian case by a $ML$ term that was barely observable in Einstein's time. So, you'd just do the ordinary Newtonian perturbation theory with one extra term that accounts for the coupling of the angular momentum to $\ddot r$
A modern approach would leverage the Post Newtonian Expansion around all of the solar system bodies. But that formalism didn't exist in 1916.