  Reflexive lattice polytopes play a key role in combinatorics, algebraic
geometry, physics, and other areas. One important class of lattice polytopes
are lattice simplices defining weighted projective spaces. We investigate the
question of when a reflexive weighted projective space simplex has the integer
decomposition property. 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 $m\geq
1$, 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 the Ehrhart $h^\ast$-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.
