feedback cooling of a low-loss and high-stress macroscopic SiN membrane resonator close to its quantum ground state. We use the microwave optomechanical platform, where the resonator is coupled to a microwave cavity. The experiment utilizes a Josephson travelling wave parametric amplifier, which is nearly quantum-limited in added noise, and is important to mitigate resonator heating due to system noise in the feedback loop. We reach a thermal phonon number as low as 1.6, which is limited primarily by microwave-induced heating. We also discuss the sideband asymmetry observed when a weak microwave tone for independent readout is applied in addition to other tones used for the cooling. The asymmetry can be qualitatively attributed to the quantum-mechanical imbalance between emission and absorption. However, we find that the observed asymmetry is only partially due to this quantum effect. In specific situations, the asymmetry is fully dominated by a cavity Kerr effect under multitone irradiation.