Soft-clamped silicon nitride membrane resonators reach coherence times tau in excess of 100 ms at milliKelvin bath temperatures. However, harnessing strong optomechanical coupling in dry dilution refrigerators remains challenging due to vibration issues and heating by optical absorption. Here, we propose to address these issues with an actuator-free optical cavity and mechanical resonator design, in which the cavity is mounted on a simple vibration-isolation platform. We observe dynamical backaction when the cavity is driven with a free-space optical beam stabilized close to the red sideband using a two-beam locking scheme. Finally, we characterize the effect of absorption heating on the coherence time, and find a scaling with the intracavity power P as tau proportional to P to the power of -(0.34+/-0.04).