In exactly the same way. One simply computes the GR prediction for the same experiment. For this, of course, the Newtonian approximation of GR will be sufficient, but you can, of course, also use full GR to compute it. Factors of 2$\pi$ appear in mathematics all the time, and some more factors 2 are nothing one would care about. If it would be possible to get rid of them by some convention, similar to "we use units with $c,\hbar=1$", mathematicians would be happy to do it. But a "we use units with $2\pi=1$" makes not really sense.