Is this closed-form summation a special case of known Lerch zeta function formulas?

This is the Fourier series for the RHS, as a function of $\alpha\in (0,2\pi)$, $$ f(\alpha)=\frac{2\pi i}{1-e^{-2\pi iz}}\, e^{-iz\alpha} . $$ The series representation follows by computing the Fourier coefficients and noting that $f$ (as a function on the circle) is smooth away from $\alpha\equiv 0\bmod 2\pi$, so the Fourier series converges to the function.

The $z$-derivative of your sum is given in M.Engelhardt and B.Schreiber, Z.Phys.A 351 (1995) 71, cf. eqs. (6) and (7) therein (sorry, I don't have a freely accessible online link). As already mentioned as a comment to Christian Remling's answer, these sums can be evaluated by pulling out a factor $e^{-iz\alpha }$ and replacing factors $1/(z+n)$ with integrations, upon which one can perform the sum.