The mistake is that $dr ds$ is exact. Using smooth forms is a bit weird to me, but not wrong. Let's use holomorphic forms, and pick an affine chart so the equation of our elliptic curve is $y^2=x^3+ax+b$ and $D$ is the point at infinity. The unique-up-to-scalar global 1-form is $y^{-1} dx$ in this chart. Then $y dx$ is a form with logarithmic singularities along $D$ and its real derivative should be the global 2-form up to a scalar.

EDIT: So my attempt at manipulating forms was a failure, so let's just go by general non-sense.

These are CDGA's in the $j^{-1}\mathcal{O}_{X}$-module category (after pulling back $A^\bullet_X(\log D))$ ), so we really just need to check that the quasi-isomorphism preserves the multiplication structure. This can be checked locally, but in local coordinates it's more-or-less obvious that the wedge of logarithmic forms maps to the wedge of the forms restricted to $X \setminus D$.