the original charge $e$ require some repackaging of the higher order effects into a revised definition of the original charge. Logically this is conceivable by renormalizing the coupling at a higher energy scale. The standard effective field theory treatment is equivalent to renormalzing the operators near zero energy. However, by choosing a renomalization condition that defines the charge at a higher energy scale $E$, one would in general include some of the higher momentum dependence into the definition of the charge. For example, for a given amplitude $Amp_i$ we would define $$\begin{aligned}
Amp_i &=& a_i g^2 +b_ig^2\kappa^2 q^2 \nonumber \\
&=& a_i g^2 (1+ \frac{b_i}{a_i}\kappa^2 E^2) + b_i g^2\kappa^2(q^2-E^2)\nonumber \\
&=& a_i g^2(E) + b_ig^2(E) \kappa^2(q^2-E^2)\,,\end{aligned}$$ when renormalizing at the scale $q^2 =+E^2$. Here $G=\kappa^2/(32\pi)$ and $a_i~b_i$ are process dependent constants. This can certainly be done by explicit construction - indeed it can be done in multiple ways through different choices of the renormalization condition. By construction, this provides a gravitational contribution to the running of the coupling with energy.

However there are obstructions that will in general keep any such construction from being useful and universal in effective field theories such as general relativity. One is a kinematic ambiguity. 