For many synchrotron radiation experiments, it is critical to perform continuous, real-time monitoring of the X-ray flux for normalization and stabilization purposes. Traditional transmission-mode monitors included metal mesh foils and ionization chambers, which suffered from low signal stability and size constraints. Solid-state detectors are now considered superior alternatives for many applications, offering appealing features like compactness and signal stability. However, silicon-based detectors suffer from poor radiation resistance, and diamond detectors are limited in scalability and are expensive to produce. Silicon carbide (SiC) has recently emerged as an alternative to both materials, offering a high-quality mature semiconductor with high thermal conductivity and radiation hardness. This study focuses on a systematic exploration of the SiC 'free-standing membrane' devices developed by SenSiC GmbH. In particular, we performed in-depth sensor-response analysis with photon energies ranging from tender (1.75 keV) to hard (10 keV) X-rays at the Four-Crystal Monochromator beamline in the PTB laboratory at the synchrotron radiation facility BESSY II, studying uniformity of transmission and responsivity compared with the state-of-the-art beam monitors. Furthermore, we theoretically evaluated the expected signal in different regions of the sensors, also taking into account the effect of charge diffusion from the SiC substrate in the case of the not-carved region.