1. Don't push the cleaning brush through the pipes and risk spiking the inside. Instead take a tip from a woodwind instrument cleaner:
Take the cleaning brush:
and fit the brush instead of the cloth:
That allows you to drop the weight right through all of the pipes including the long bendy one and pull the brush through keeping its spike away from the rubber. I would tend to spray the brush with carb cleaner first!!
2. Instead of the oil drip bowl, use a baby bath which will accomodate the entire rocker cover. Then submerge the cover completely and shake in your choice of solvents, hot detergent, plastic safe degreaser etc. That avoids all the messing about with the pump. Rinse off the same way until there is just clean water in the bath.
3. Dry in a warm environment, with airflow through the cover.
Good luck with that!