There are multiple books about ways to characterize the normal distribution. Eg, Bryc’s [book][1] 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.


  [1]: https://homepages.uc.edu/~brycwz/probab/charakt/charakt.pdf