If I have an elliptic function $f$ with respect to some lattice $L$, and $f$ has poles in the points of $L$ and nowhere else, does it follow that $f$ is even?
I think so, and my idea was to write $f = S(\wp) + T(\wp)\wp'$ with rational functions $S$ and $T$, noting that $\wp$ is even and satisfies this pole condition, and $\wp'$ does not. It doesn't seem to work, though.
Maybe there's an easier way?
On
This doesn't follow, as your original claim that $\wp'$ doesn't satisfy this condition is false. We have $$ \wp'(z) = -2\sum_{\omega\in L}\frac{1}{\left(z - \omega\right)^3}, $$ which clearly has poles at the latice points (and only the lattice points by standard convergence considerations). And, as you said, $\wp'$ is odd.
$\wp^\prime$ has poles only at the lattice points, and is known to be odd, so...