It seems like there is an upper bound on how many words one can use in a comment. Let me explain a little more. Basically I need an isomorphism between $H^i(Ext(\Omega_D, \mathcal{O}_D)$ to $H^{i+2}(Hom(\Omega_D, \mathcal{O}_D))$. One way to do this is to try to relate these to the ambient weighted projective space.

Well, it is kind of a long story why I am interested in $\Omega_X$ rather than its reflexsive hull. Roughly speaking because the deformation theory is controlled by $\Omega_X$. I am trying to understand the deformation of a pair $(X, D)$. A question I asked a few days ago is here: mathoverflow.net/questions/45343/… The divisor $D$ is isomorphic to a (rather singular, but normal) weighted hypersurface in a weighted projective space. (But $X$ is not the weighted projective space!) So I am trying to relate the cohomology of D to the w.p.s.