We present a recent Atacama Large Millimeter/submillimeter Array observation of the CO(1-0) line emission in the central galaxy of the Zw 3146 galaxy cluster (z = 0.2906). We also present updated X-ray cavity measurements from archival Chandra observations. The 5 × 1010 M⊙ supply of molecular gas, which is confined to the central 4 kpc, is marginally resolved into three extensions that are reminiscent of the filaments observed in similar systems. No velocity structure that would be indicative of ordered motion is observed. The three molecular extensions all trail X-ray cavities, and are potentially formed from the condensation of intracluster gas lifted in the wakes of the rising bubbles. Many cycles of feedback would be required to account for the entire molecular gas reservoir. The molecular gas and continuum source are mutually offset by 2.6 kpc, with no detected line emission coincident with the continuum source. It is the molecular gas, not the continuum source, that lies at the gravitational center of the brightest cluster galaxy. As the brightest cluster galaxy contains possible tidal features, the displaced continuum source may correspond to the nucleus of a merging galaxy. We also discuss the possibility that a gravitational wave recoil following a black hole merger may account for the displacement.