In view of my answer here, it is highly unlikely that the distribution in question can be obtained in closed form. However, it is not hard to show that this distribution is approximately normal. Indeed, the distribution in question is the distribution of the random variable (r.v.)
where $(U,V)$ is the sum of
iid copies of the random pair
where $X_1,X_2,Y_1,Y_2$ are iid $\mathcal N(0,\si^2/2)$. The mean of the pair $(A,B)$ is $(\si^2,0)$ and its covariance matrix is diagonal with $5\si^4/2$ and $\si^4/2$ on the diagonal. So, by the multivariate (here bivariate) central limit theorem, the joint distribution of $(U,V)$ is approximately the bivariate normal distribution
with means $M\si^2$ and $0$, variances $5M\si^4/2$ and $M\si^4/2$, and zero correlation.
So, if $M$ is large, then, by the multivariate delta method (with $h(u,v)=u^2+v^2$), the distribution in question, which is the distribution of $U^2+V^2$, is