There is no mirror object at L3 because we have sailed to the "dark side" of the Sun with more than one probe (e.g. Parker), and nada.
All the Earth's La Grange points are too close to the Sun to help find a distant planet in our system. Also, at 500 AU, the orbital period is over 11,000 years, so catching perturbations in orbits of, say, Neptune might be a problem.
The inability to see large planets just a few thousand AU away, while being able to see these same -sized objects that are a million or more AU away orbiting other stars is one of those cute quirks I find that add to the greatness of astronomy.
Consider that Jupiter is invisible to the HST if it were to move out to only ~ 10,000 AU, 1/6th the distance to the outer solar system. [The JWST, however, can see it farther out because of its larger mirror and its IR capabilities.] But both of these can observe directly a select few orbiting other star systems.
The reason is the inverse fourth power, which is the inverse square law for light propagating out of the Sun. But it is also the inverse square law that must be applied to the light that is reflected off the planet, hence their product (inv. 4th power). Not many photons make it back.