Preceding answer suppressed because wrong. New version coming soon.

EDIT. New answer.

Due to quantum corrections, the propagator of the photon is some multiple $(1+F(M))$ of the freestandard propagator. At the one loop order, we have $F(M) = e(M)^{2}G(M)$ for some function $G$. The free standard propagator of the photon in $1/p^{2}$ gives rise to a static potential between two electrons of charges $e$ separated by a large distance $r$ in $e^{2}/r$, this is the standard Coulomb law. Given the quantum corrections to the photon propagator, this static potential at long distances is modified in $e(M)^{2}(1+F(M))/r$. This shows that the function $e(M)^{2}(1+F(M))$ is a physical quantity which can be experimentally measured. In particular, it can not depend of the renormalization scale $M$ i.e. $\frac{d}{dM}(e(M)^{2}(1+F(M)))=0$.

At the one loop order, this implies

$\beta(e(M)) = M \frac{d}{dM} e(M) = - \frac{e(M)}{2} M \frac{d}{dM}F(M)$.