For the single-rope pickoffs I have done, the rescuer is on ascenders, not rappel device. It does not matter whether the rescuer starts above or below the patient. The rescuer will need to get above the patient in either case. This can be accomplished by climbing up past the patient, or downclimbing on ascenders from above. As most cave trips are not pull-downs, we are usually ascending up past the patient on the patient's line. (It is the only line.)
One step I left out for clarity earlier but feel obligated now to mention, is that you clip a safety tether between rescuer and patient as soon as patient is reached. This allows for removal of patient's gear from the rope in order to clean up the situation and prepare for lower.
If you read my post above, you will see I am talking about lowering the patient from a rap device attached to the rescuer's ascenders. This takes care of the weighted rope below you problem.
Once again, pickoffs are a last resort answer. Consider another course of action first.