I don't know these new systems, but could you not just wind the cabin alt up a bit? I expect it would be below 8000ft.
You can't set the cabin altitude specifically on that system. Before take off you tell the computer your cruise altitude and landing elevation. It works out the rest. I am thinking the OP could set a lower cruise altitude in the window. Below FL370 the max diff pressure is reduced slightly. That might relief some of the excess pressure inside. If so you'll end up with a higher cabin altitude. This is speculation on my part though, not sure if that is what will happen for real...