  We developed a numerical methodology to compute the fully-relativistic
propagation time of photons emitted by a pulsar in orbit around a massive
compact object, like the supermassive black hole Sagittarius A* in the Galactic
Center, whose gravitational field is described by a generic spherically
symmetric space-time. Pulsars at the Galactic Center are usually regarded as
the next major precision probe for theories of gravity, filling the current
experimental gap between horizon-scale gravity tests and those at larger
scales. We retain a completely general approach, which allows us to apply our
code to the Schwarzschild space-time (by which we successfully validate our
methodology) and to three different well-motivated alternatives to the standard
black hole paradigm. The results of our calculations highlight departures
spanning several orders of magnitudes in timing residuals, that are supposed to
be detectable with future observing facilities like the Square Kilometer Array.
