The question provides asymptotic results as $n$ becomes large, but does not state whether your focus of interest is restricted to large $n$, or small $n$, or for any $n$.

I would suggest you change the title from 'moment' to 'fractional moment'. The term $r^{th}$ moment is conventionally taken to refer to integer values of $r$.

Do you wish to assume than the $X_i$ are not only independent, but also identically distributed? Because that is currently missing.

Why is this interesting? Very messy question.

So where is the parameter $\alpha$ in your definition? And if it does not even appear in your definition, how do you propose to impose the constraint $0 < \alpha \leq 2$.

Stable distributions are essentially defined by their characteristic function, so to request a proof that side-steps the cf (Fourier transform) seems a bit unreasonable.

Actually, explicitly ... it nests all the requirements that the marginal distributions of $X$ and $Y$ are standard Normal as well as any desired correlation $\rho$ between $X$ and $Y$, ... though there would be, of course, alternative models. The real point is to illustrate that the maximum bound will be a function of $\rho$, ... whereas to merely describe it as $\sqrt{\frac{2}{\pi}}$ is a bit of a broad sword that lacks proper aim.