This is a divergent series. But if one applies summation in the sense of Eisenstein,
then the sum is doubly periodic. Since the poles are at the lattice and residues are equal to $1$, it is equal $\wp(z)+C$. Looking at the Laurent expansion at $0$ we obtain $C=0$. So your sum is the Weierstrass function (if it is understood in the sense of Eisenstein).
Remark. The inner sum in parentheses is absolutely convergent.
Ref. A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer, 1976.
For Eisenstein summation, see also
Remmert, Classical topics in complex function theory, Springer 1998. He uses it to define trigonometric functions.