Bianisotropic metasurfaces empower planarized optical components with subwavelength thicknesses that are not amenable to conventional optics. In particular, bianisotropy gives control over the spatial dispersion and allows engineering of both spectral and directional responses simultaneously, yielding devices with unprecedented functionalities. In this work, we demonstrate a nonlinear bianisotropic metasurface that features asymmetric second-harmonic generation (SHG). The concept of the device is as follows: SHG from the metasurface occurs only upon excitation in the forward direction, whereas backward excitation does not result in efficient SHG, effectively making the metasurface perform as a nonlinear pseudo-diode. Such functionality is achieved by employing two common plasmonic materials – aluminium (Al) and silver (Ag) as constitutive materials of the meta-atos. Apart from exploiting the stronger second- order nonlinear response of Al, we show that tailored mixing of these two metals provides an elegant and simple approach to engineer the spatial dispersion, which is required to achieve such an asymmetric response. In turn, the geometrical parameters of the metasurface are optimized to exhibit SHG maximum upon excitation at 800 nm wavelength (resulting in SHG peak at 400 nm). We demonstrate the experimental implementation of the metasurface and discuss the fabrication method that is based on a bottom-up approach. To further substantiate our findings, we perform a homogenization analysis, which allows for extracting the effective susceptibility tensors and unambiguously reveals the bianisotropic response of our metasurface. Finally, we discuss the implications of our results from the more fundamental perspectives of reciprocity and time-reversal asymmetry.