Can we detect the wobble of neighbouring stars around their barycenters?
Yes it is. The Gaia astrometric satellite is and will be doing exactly that. The subtle movement of the photocentre of a target star due to any orbiting planets can be separated from the proper motion and the annual parallax due to our motion around the Sun.

The technique is most sensitive to massive planets that are widely separated from  parent stars that are close to the Sun and have low mass themselves, since these provide the biggest angular shift in the star's position with respect to the barycentre. However, the separations cannot be too wide or the orbital period of the star/planet system will be too long for the motion to be detectable on the timescales of the mission. In practice therefore, Gaia will be most sensitive to planets with orbital periods of 1-10 years (or separations of 1-5 au). Simulations suggest that tens of thousands of exoplanets may be revealed in this way (Perryman et al. 2014).

ESA have even provided a little animation to show how it works.