There are multiple books about ways to characterize the normal distribution. Eg, Bryc’s book starts with Herschel-Maxwell’s
Theorem: If $X$ and $Y$ are independent variables whose joint distribution is rotationally invariant, then $X$ and $Y$ are both normal.
He immediately notes that one can strengthen this to Polya’s:
Theorem: If $X$ and $Y$ are independent variables, and rotations of $\pi/4$ and $\pi/2$ leave the distribution of $X$ invariant, then $X$ and $Y$ are both normal.
Perhaps somewhere in such books you’ll find a characterization with hypotheses that avoid moments but are number-theoretically tractable.