Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice sim-plices defining weighted projective spaces. We investigate the question of when a reflexive weighted projective space simplex has the integer decomposition prop-erty. We provide a complete classification of reflexive weighted projective space simplices having the integer decomposition property for the case when there are at most three distinct non-unit weights, and conjecture a general classification for an arbitrary number of distinct non-unit weights. Further, for any weighted projective space simplex and m ≥ 1, we define the m-th reflexive stabilization, a reflexive weighted projective space simplex. We prove that when m is 2 or greater, reflexive stabilizations do not have the integer decomposition property. We also prove that as long as one weight is at least three, the Ehrhart h*-polynomial of any sufficiently large reflexive stabilization is not unimodal and has only 1 and 2 as coefficients. We use this construction to generate interesting examples of reflexive weighted projective space simplices that are near the boundary of both h*-unimodality and the integer decomposition property.