Since G is finite, there is no problem with just repeating the proof in the 
case G=e, using Z-graded homotopy group functors on the orbit category. Take
D^{\leq n} to be the spectra whose homotopy groups \pi_q(X^H) are zero for q>n, 
and dually for D^{\geq n}.  The intersection for n\leq 0 and n\geq 0 consists 
of the Eilenberg-MacLane G-spectra K(M,0) for Mackey functors M.   Peter May