Hi Chris.....Vickie B. here, wife of Roy who has been into bees for just 2
seasons now.  I  had to chuckle at your request for a way to remove the bee
stains!  I'm "with your wife" on that one....but Roy just washes his as is,
and of course, the stains don't come out.  Since I work on the "sidelines" I
don't have my own bee suit, but if I did I'd try to bleach it I guess.
 
I would imagine, however, that you will get some good suggestions from some
experienced beekeepers. However, it could be a "badge of honor" thing, kind
of like the Hells Angels Bikers' jackets, the dirtier the better!!
 
Now then, how does your wife cope with the "seasonal" sticky doorknobs and
counters?  I've just learned to smile sweetly and wipe.  Roy loves his
beekeeping so much....I just can't complain any more.  :+)