We consider lattice simplices corresponding to weighted projective spaces where one of the weights is 1. We study the integer decomposition property and Ehrhart unimodality for such simplices by focusing on restrictions regarding the multiplicity of each weight. We introduce a necessary condition for when a simplex satisfies the integer decomposition property, and classify those simplices that satisfy it in the case where there are no more than three distinct weights. We also introduce the notion of reflexive stabilizations of a simplex of this type, and show that higher-order reflexive stabilizations fail to be Ehrhart unimodal and fail to have the integer decomposition property. This is joint work with Robert Davis, Derek Hanely, Morgan Lane, and Liam Solus.