Ultraintense laser–plasma experiments generate a variety of high-energy radiations, including nonlinear inverse Compton scattered (NCS) X-rays, which are expected to be a key experimental observable as we transition into the quantum electrodynamic plasma regime. However, there is also a high bremsstrahlung X-ray background that reduces our ability to observe NCS X-rays. Previous numerical studies comparing NCS and bremsstrahlung emissions fail to capture the full temporal emission of both processes. We present for the first time two-dimensional particle-in-cell (PIC) and three-dimensional hybrid-PIC EPOCH simulations that capture up to 150 ps of the laser–plasma interaction and directly compare the NCS and bremsstrahlung emissions for a plastic target for intensities of ${10}^{20}{-}{10}^{23}$ W/cm2. We present angular distribution plots where the NCS emission is seen to dominate at intensities greater than 5$\times {10}^{21}$ W/cm2 and the target design is seen to successfully divert the bremsstrahlung signal away from the NCS lobe regions, making the experimental observation of nonlinear inverse Compton scattering at lower intensities more likely.