I think you could probably get away with not washing, even if you don't have room in your cooler. Breastmilk is good on the counter at room temp for 5 hours -- do you go 5 hours between pumping? I say, pump, put the parts in a clean bag, pump the second time and then wash when you get home. Does that make sense? So maybe some 3-hour old milk from the pump winds up in the bottle when you pump the second time, but that's still okay -- because that milk then goes straight into the cooler and the five or eight or however old milk it is on the pump by the time you get home all gets washed off anyway.
I only pump once at work, and I can refigerate the parts, so that's just an idea, not something I have experience with. Good luck! (And I also sometimes pump in a closet! But then I walk the milk down to the lounge with a fridge in it.)
, mama to Amelie (May 2010), early loss (October 2011), and James (September 2012)