Carbon nanotubes are excellent nanoelectromechanical systems, combining high resonance frequency, low mass, and large zero-point motion. At cryogenic temperatures they display high mechanical quality factors. Equally they are outstanding single-electron devices with well-known quantum levels and have been proposed for the implementation of charge or spin qubits. However, the integration of these devices into microwave optomechanical circuits is hindered by a mismatch of scales between typical microwave wavelengths, nanotube segment lengths, and nanotube deflections. As experimentally demonstrated recently by Blien et al. [Nat. Comm. 11, 1363 (2020)], coupling enhancement via the quantum capacitance allows this restriction to be circumvented. Here we extend the discussion of this experiment. We present the subsystems of the device and their interactions in detail. An alternative approach to the optomechanical coupling is presented, allowing the mechanical zero-point motion scale to be estimated. Further, the mechanical damping is discussed, hinting at hitherto unknown interaction mechanisms.