Could the barycenter orbit of our sun be greatly underestimated? The radius of the suns orbit around the barycenter of the solar system is often measured to be roughly the radius of the sun. Based on the same type of bias that has made humans miss things about their solar system because they are looking at it from the vantage point of Earth orbiting inside the system, is it possible that the suns orbit around the barycenter is actually much larger?
This is basically equivalent to asking whether there's an undetected massive object in the outer Solar System.

Starting from the basics, the location of the centre-of-mass of a system of point masses is given by

$$\mathbf{X} = \frac{\sum_{i} m_{i} \mathbf{x}_{i}}{\sum_{i}m_{i}}$$

Note that the quantities in bold are vector quantities. Without loss of generality, we can use a reference frame where $\mathbf{X} = \mathbf{0}$, i.e. the origin of the coordinate system is the centre-of-mass. For a two-body system, this gives:

$$m_{1} \mathbf{x}_{1} = -m_{2} \mathbf{x}_{2}$$

So the second particle is located in the opposite direction of the first from the centre-of-mass, with the ratio of the distances $r_{1} = \left\| \mathbf{x}_{1} \right\|$, $r_{2} = \left\| \mathbf{x}_{2} \right\|$ depending on the mass ratio $q = m_{2} / m_{1}$:

$$r_{1} = qr_{2} = \left( \frac{q}{1+q} \right) r$$

Where $r = r_{1} + r_{2}$ is the total separation between the two objects. This means that the trajectories of the two objects around the centre-of-mass are scaled mirror-images of each other. So the semimajor axis of the reflex orbit of the Sun $a_{1}$ for a planet of mass $m_{2}$ with orbital semimajor axis $a$ is given by:

$$a_{1} = qa_{2} = \left( \frac{q}{1+q} \right) a$$

This orbit is larger for more massive planets (higher $q$) and for wider planetary orbits (higher $a$). Working this out for the major planets, the largest reflex orbit is due to Jupiter (1.06 solar radii), followed by Saturn (0.59 solar radii), Neptune (0.33 solar radii) and Uranus (0.18 solar radii).

If Planet Nine exists with the predicted properties given by Batygin et al. (2019), a ~5–10 Earth mass planet in a ~400–800 AU orbit would give a reflex orbit of ~1 to ~5 solar radii, which would likely be the largest contribution to the Sun's orbit around the Solar System barycentre.

The hypothesis that the Sun may have a distant binary companion or a giant planet in the Oort cloud (which would both be more massive than the proposed Planet Nine, and much further out), which has variously been termed "Nemesis" or "Tyche" would result in an even larger orbit around the Solar System barycentre. So far, these hypotheses have not held up to observational evidence.