What you want is not possible. Suppose $$n>3$$, and consider making $$A_2=\sum_{i<j}(X_iX_j+Y_iY_j)$$. All polynomials available are homogeneous in the total degree, and the degree of the $$C_i$$ is too high to be of any use, so you'll have to make do with the $$B_i$$. But the only way to get a monomial $$X_iX_j$$ is to multiply $$B_iB_j$$; however that gives you mixed terms $$X_iY_j$$ and $$X_jY_i$$ as well, which you don't want. There is no way to get only the terms in $$A_2$$.