Radar-based sensing is a promising privacy-preserving alternative to cameras and wearables in settings such as long-term care. Yet detecting quasi-static presence (lying, sitting, or standing with only subtle micro-motions) is difficult for low-resolution SIMO FMCW radar because near-zero Doppler energy is often buried under static clutter. We present Respiratory-Amplification Semi-Static Occupancy (RASSO), an invertible Doppler-domain non-linear remapping that densifies the slow-time FFT (Doppler) grid around 0 m/s before adaptive Capon beamforming. The resulting range-azimuth (RA) maps exhibit higher effective SNR, sharper target peaks, and lower background variance, making thresholding and learning more reliable. On a real nursing-home dataset collected with a short-range 1Tx-3Rx radar, RASSO-RA improves classical detection performance, achieving AUC = 0.981 and recall = 0.920/0.947 at FAR = 1%/5%, outperforming conventional Capon processing and a recent baseline. RASSO-RA also benefits data-driven models: a frame-based CNN reaches 95-99% accuracy and a sequence-based CNN-LSTM reaches 99.4-99.6% accuracy across subjects. A paired session-level bootstrap test confirms statistically significant macro-F1 gains of 2.6-3.6 points (95% confidence intervals above zero) over the non-warped pipeline. These results show that simple Doppler-domain warping before spatial processing can materially improve semi-static occupancy detection with low-resolution radar in real clinical environments.