I suppose you can just randomly perturb the independent distribution. That is, let $$\mu(x_i,y_j)=\mu_x(x_i)\mu_y(y_j)+\epsilon_{i,j}$$ where the $\epsilon_{i,j}$ form a random matrix (sufficiently random), with the conditions that $$\sum_i\epsilon_{i,j}=0=\sum_{j}\epsilon_{i,j}\qquad \forall i,j$$ (all columns and rows add to 0), $$0\le \mu(x_i,y_j)\le 1\qquad \forall i,j$$ (entries $\epsilon_{i,j}$ are sufficiently small).

Not sure how to carry that out but there is some literature such as Barvinok: Matrices with prescribed row and column sums.

I suppose you can just randomly perturb the independent distribution. That is, let $$\mu(x_i,y_j)=\mu_x(x_i)\mu_y(y_j)+\epsilon_{i,j}$$ where the $\epsilon_{i,j}$ form a random matrix (sufficiently random), with the conditions that $$\sum_i\epsilon_{i,j}=0=\sum_{j}\epsilon_{i,j}\qquad \forall i,j$$ (all columns and rows add to 0), $$0\le \mu(x_i,y_j)\le 1\qquad \forall i,j$$ (entries $\epsilon_{i,j}$ are sufficiently small).

In theory you could pick these randomly with respect to Lebesgue measure on the set of all such matrices.