Ok so the solution is not to consider the product of ideals instead of their intersection.

Just like we have $$ B_{\widetilde{I}_1} B_{I_2} X = B_{I_1I_2} X = B_{\widetilde{I}_2} B_{I_1} X $$ with $\widetilde{I}_j$ the total transform of the ideal $I_j \subset O_X$, we have $$ D_{\widetilde{D}_1} D_{A_2} X = D_{(A_1,A_2)} X = D_{\widetilde{D}_2} D_{A_1} X $$ with $\widetilde{D}_j = A_i \times_X D_{A_j} X$ the total pullback of the normal cone deformation space to $A_i$.

Ok so the solution is to consider the product of ideals instead of their intersection.

Just like we have $$ B_{\widetilde{I}_1} B_{I_2} X = B_{I_1I_2} X = B_{\widetilde{I}_2} B_{I_1} X $$ with $\widetilde{I}_j$ the total transform of the ideal $I_j \subset O_X$, we have $$ D_{\widetilde{D}_1} D_{A_2} X = D_{(A_1,A_2)} X = D_{\widetilde{D}_2} D_{A_1} X $$ with $\widetilde{D}_j = A_i \times_X D_{A_j} X$ the total pullback of the normal cone deformation space to $A_i$.

With this definition, we have nice functoriality properties w/r to maps and direct products.