In mid-2012, a global merged interaction region (GMIR) observed by Voyager 2 crossed through the heliosheath and collided with the heliopause, generating a pressure pulse that propagated into the very local interstellar medium. The effects of the transmitted wave were seen by Voyager 1 just 93 days after its own heliopause crossing. The passage of the transient was accompanied by long-lasting decreases in Galactic cosmic ray intensities that occurred from ∼2012.55 to ∼2013.35 and ∼2012.91 to ∼2013.70 at Voyager 2 and Voyager 1, respectively. Omnidirectional (>20 MeV) proton-dominated measurements from each spacecraft’s Cosmic Ray Subsystem reveal a remarkable similarity between these causally related events, with a correlation coefficient of 91.2% and a time lag of 130 days. Knowing the locations of the two spacecraft, we use the observed time delay to calculate the GMIR’s average speed through the heliosheath (inside the heliopause) as a function of temperature in the very local interstellar medium. This, combined with particle, field, and plasma observations, enables us to infer previously unmeasured properties of the heliosheath, including a range of sound speeds and total effective pressures. For a nominal temperature of ∼20,000 K just outside the heliopause, we find a sound speed of 314 ± 32 km/s and total effective pressure of 267 ± 55 fPa inside the heliopause. We compare these results with the Interstellar Boundary Explorer’s data-driven models of heliosheath pressures derived from energetic neutral atom fluxes (the globally distributed flux) and present them as additional evidence that the heliosheath’s dynamics are driven by suprathermal energetic processes.