Biological cells are capable of undergoing extensive shape transformations thanks to the existence of membrane area reservoirs from which they can pull out membrane when required. A particularly relevant example of such membrane remodelling is given by endocytic and phagocytic processes, during which the cell membrane engulfs nano- and micrometer sized particles. Recently, it was shown that cell-like membrane reservoirs can be mimicked in giant vesicles with nanotubes stabilized by strong bilayer asymmetry, as quantified by the membrane's spontaneous curvature. Here, we theoretically investigate particle engulfment by such strongly-asymmetric membranes. We find that, depending on the sign of the spontaneous curvature, the engulfment transition may be continuous or discontinuous. Moreover, we find that, in the case of particle engulfment, the presence of asymmetry-stabilized reservoirs is not well captured by the constant-tension model typically used to describe cell-membrane deformations. This highlights the need for a better understanding of the nature of cellular membrane reservoirs, in order to accurately describe membrane remodelling processes.