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.