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Mechanical feedback from active galactic nuclei (AGN) is thought to be the dominant feedback mechanism quenching cooling flows and star formation in galaxy cluster cores. However, the mechanisms by which AGN couple to the intracluster medium (ICM) are not well understood. The nature of pressure supporting the cavities is not known. Using the MUSTANG-2 instrument on the Green Bank Telescope (GBT), we aimed to measure thermal Sunyaev-Zeldovich (SZ) effect signals associated with the X-ray cavities in MS0735.6+7421, a moderate mass cluster hosting one of the most energetic AGN outbursts known. We use these measurements to infer the level of non-thermal sources of pressure, such as magnetic fields and turbulence, as well as relativistic and cosmic ray components, supporting the cavities. We used preconditioned gradient descent to fit a model for the cluster, cavities, and central point source directly to the time ordered data of the MUSTANG-2 signal. We use this model to probe the thermodynamic state of the cavities. We have shown that the SZ signal associated with the cavities is suppressed compared to the expectations for a thermal plasma with the temperature $\sim$few tens keV. The smallest value of the suppression factor $f$ that is consistent with the data is $\sim$0.4, lower than inferred in earlier work. Larger values of $f$ are possible once the contribution of the cocoon shock surrounding the bubbles is taken into account. The baseline model with this particular geometrical setup yields best-fitting value f~0.5, which at face value implies a mix of thermal and non-thermal pressure support. Larger values of $f$ (up to 1, i.e. no tSZ signal from the bubbles) are still possible when allowing for variations in the line-of-sight geometry.