Can something be said when all (X_i) and (Y_i) are assumed to be independent?

I would try as compensator for the Poisson random measure $\nu(dy) = \delta_0(d y_1) \nu_2(d y_2) + \nu_1(d y_1) \delta_0(d y_2)$. The drift is essentially $b = (b_1, b_2)$, corrected for the changed unit ball ( $1_{\{|y_1| \leq 1\}} 1_{\{|y_2| \leq 1 \}}$ which becomes $1_{\{ ||(y_1, y_2)|| \leq 1 \}}$). Does this work out for you?