I'm one who made a snide comment about hand washing - mostly in jest, but obviously your situation requires it.... unless you buy two weeks worth of cycling attire so you can save it up for your trips to the laundromat.
As for the washing board, my opinion is that using a washing board would be hard on the synthetic clothes, as hard or harder than using a normal cycle washing machine. A washing board is for hard scrubbing for stuff that's durable and really dirty. Gentle hand washing technique has always avoided the washing board and just used soaking, hand squeezing, scrubbing material against material, or "kneading" on a smooth surface. By the way, soaking for a while will do 90% of the work for you! At least that's what my mom did. She just used the washing board (built into her large laundry sink) for the heavy cotton and/or really dirty stuff. This was a divided deep, two sink set up. Both sinks had kind of angled front side for washing clothes. One sink was smooth, the other sink had a built in washing board. Yes, I did some time handwashing under mom's supervision. She'd sit on a chair smoking a cigarette and nursing a cocktail yelling at us kids to scrub harder and beat us if the clothes weren't perfect!