= Solution
Set $a=\Sigma_{11}$, $c=\Sigma_{12}$, and $b=\Sigma_{22}$. The estimator is the sample mean of the independent random vectors
$$
W_i=(X_i^2,X_iY_i,Y_i^2)^T,
\qquad
\mathbb EW_i=(a,c,b)^T.
$$
The supplied fourth moments give
$$
\begin{aligned}
\operatorname{Var}(X^2)&=2a^2,&
\operatorname{Cov}(X^2,XY)&=2ac,&
\operatorname{Cov}(X^2,Y^2)&=2c^2,\\
\operatorname{Var}(XY)&=ab+c^2,&
\operatorname{Cov}(XY,Y^2)&=2bc,&
\operatorname{Var}(Y^2)&=2b^2.
\end{aligned}
$$
Therefore the <multivariate central limit theorem> gives
$$
\boxed{
\sqrt n
\left[
\begin{pmatrix}
\widehat\Sigma_{11}\\
\widehat\Sigma_{12}\\
\widehat\Sigma_{22}
\end{pmatrix}
-
\begin{pmatrix}a\\c\\b\end{pmatrix}
\right]
\xrightarrow d
N_3\!\left(0,
\begin{pmatrix}
2a^2&2ac&2c^2\\
2ac&ab+c^2&2bc\\
2c^2&2bc&2b^2
\end{pmatrix}\right).}
$$
This is the <asymptotic covariance of the bivariate Gaussian covariance estimator>.
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