Are you planning to retain the drywell? Eventually, the ground gets plugged up with lint and soap scum and can't absorb water at a decent rate, plus, in some places, it's illegal.
But, that isn't a p-trap, it's an S-trap. You may or may not need a vent line to the drywell, but having it terminate in the same room is just a waste.
My mother's house had a similar situation. WHat I ended up doing was installing a Liberty pump designed specifically for this application and running it into the main line to the septic system. The pump is small enough that it fit right beside the sink. We ran the outlet of the WM into the sink, and the sink drained into the tank of the pump, then the pump's float switch turned on and pumped it to the drain.