The measurements discussed weren't done by Messenger, they were done by round-trip delay-doppler radar ranging using the coherent transponder of Messenger's Radio_science_subsystem.
Park et al. (2017) discuss separating the effects of the solar quadrupole moment from the partially degenerate effects of the post-Newtonian $\beta$ and $\gamma$ parameters (where $\beta=\gamma=1$ in General Relativity). The "general" General Relativistic correction to the perhelion prececssion itself is given by:
$$\Delta \dot{\omega} \propto (2 - \beta + 2\gamma) $$
and the effect caused by the solar quadrupole moment is only 0.07% the size of this. Therefore just observing the precession does not yield an accurate value of $J_2$ without assuming the value of $\beta$ and $\gamma$, and even then would need precisons in position measurements of hundreds of metres over decades.
Park et al. instead show that separating $J_2$ from $\beta$ and $\gamma$
requires a precise measurement of a periodic modulation (i.e. it isn't an accumulating error) in the argument of perhelion caused by the General Relativistic perturbations, which translates to modulations of order of 5-10 km in position. These also depend linearly on $\beta$ and $\gamma$ only, and if the amplitudes of these modulations can be measured to better than $\sim 0.07$% ($\simeq 3.5-7$ m) then $J_2$ could be determined independently of $\beta$ and $\gamma$.
Given that the position of the spacecraft can be tracked to a lttle better than 1 metre, this is possible. $J_2$ is indeed measured to about 4%, and half of this uncertainty is due to remaining degeneracy with the $\beta$ parameter and uncertainty in the adopted $\gamma$ parameter.
This suggests that precisions of the order of a few metres were required for this experiment and I don't believe this is possible by radar ranging to the planet itself.
Indeed on p.181 of the 1993 version of "Theory and Experiment in Gravitational Physics by Will, it is said that
Unfortunately, measurements of the orbit of Mercury alone are incapable at present of separating the effects of relativistic gravity and of solar quadrupole moment in the determination of [the change in the perihelion precession rate]
and then on p.183
The accuracy required for such measurements would necessitate tracking of a spacecraft in orbit around Mercury