There's a very rich literature about these things: on one hand, there is the pure general relativity of light bending due to an oblate planet (a quadrupolar effect), and on the other hand there is the observational effort to detect such light bending in our solar system. I think the first was addressed, more or less, in my answer to one of your previous questions:
Has GAIA learned anything about General Relativity looking near Jupiter?
The question here pertains to previous observational efforts:
Question: Is GAIA the only game in town for looking at quadrupole gravitational deflection of light? Is there some other method with similar sensitivity, using either Jupiter or the Sun which is less oblate but much more massive? Radio perhaps, somehow?
Short answer: Yes, Gaia is the only game in town currently, although there is a tradition of ground-based observations trying to observe the monopolar contribution to light bending of Jupiter (and I suspect that future ground based telescopes might be able to). Also, there was a competitor, in a sense, to Gaia, known as the Space Interferometry Mission (SIM) but it was canceled in NASA's Astro2010 Decadal Report.
For "recent/modern" astrometric observations of light bending with the Sun, one can dig into the weeds a bit:
Detection of light deflection around the Sun and the Earth was accomplished by Hipparcos (1992), see r.f. section 3.2.5. of this study by Lindl (2011), discussed more down below, which details the theory behind astrometry of light deflection via Jupiter with Gaia. The light bending observed by Hipparcos was due to the monopole moment, not the quadrupole moment, of the lens. These successes bolstered proposals to observe the quadrupolar contribution of light deflection.
On the second page of this paper by Heinkelmann and Schuh (2009), where $\gamma$ is the parameterized post-Newtonian parameter that is unity for classical general relativity, they state:
Up to now several groups have determined the $\gamma$ parameter using the geodetic VLBI observations... All of those tests focus on the effects imposed by the Sun. However, there were also several efforts to observe Jupiter’s deflection. Treuhaft & Lowe (1991) tried to find the deflection by Jupiter experimentally using a single long baseline DSN experiment during a near-occultation event, which was proposed by Schuhet al.(1988). A comparable near-occultation happened in 2002 and was investigated by several groups,e.g. by Fomalont & Kopeikin (2003).
I'm pretty sure they're referring to the monopole deflection of Jupiter here, as corraborated in a paper by Kopeikin & Makarov (2008), and not the quadrupolar deflection - an early theoretical model of which was explored by Crosta et al. 2006.
Now, some history of observing the quadrupolar contribution to light bending of Jupiter:
The Kopeikin & Makarov (2008) paper speculated that SIM, or possibly SKA, was the best prospect for measuring light bending of Jupiter, and barely mentioned Gaia.
The seminal work by Klioner (2003) presented a self-consistent relativistic model (at the 0.1 $\mu$as-level) for observing the quadrupolar light bending of Jupiter, in anticipation of Gaia and SIM, and showed that such light bending would be strongest for Jupiter (excluding the sun!), ie. see the second column of table 1 titled, "Various Gravitational Effects on Light Propagation". This model forms the base of the relativistic model of Gaia (known as GREM).
This and subsequent works motivated the study by Lind (2011) concerning this topic, where they implement the model of Crosta et al. (2006) for simulated data. From their conclusion:
The framework that we have obtained will ... give a first estimation of Gaia’s capacity to detect quadrupole light deflection by Jupiter. This we can answer in the affirmative, Gaia and AGIS will allow to detect this effect to at least 6$\sigma$ level, when assuming nominal astrometric performance.
Prior to and after the launch of Gaia in 2013, the prospects for its ability to probe fundamental physics and relativity theory was highly anticipated, for example here.
Lastly, (phew!) for a more complete authoritative source for current and upcoming microarcsecond astrometry observations is this review. Their section 6 discusses current and upcoming ground, such as the ESO's Extremely Large Telescope, and space based projects, such as NASA's James Webb Space Telescope and the Nancy Grace Roman Space Telescope and of course the ESA's Gaia.
In section 3, they state:
In particular the scanning law of Gaia has been optimized such that observations of bright stars near Jupiter are made a number of times during the mission.These data will allow an attempt at measuring the quadrupole moment of light deflection by Jupiter. How Gaia astrometry can be used in these and other tests of fundamental physics is discussed by Klioner (2014). Such tests will probably only be done in later phases of Gaia data processing operations, when the other parameter sets in the astrometric solution are sufficiently well known and understood
So these kinds of nuggets for observing general relativistic light bending with Jupiter might not arrive for a few years. But I'll certainly be keeping my ears to the ground ;)