I looked around a bit, then decided to change the preposition and see if that made a difference. Searching "beard for the washing" brought up a book called A Glossary by Robert Nares (1822). In it he has an entry for "HEAD":
"To give one's head for the washing. This very odd proverb... seems to imply, to yield tamely and without resistance, to give up your head as if it was only to be washed.
And so am I, and forty more good fellows,
That will not give their heads for the washing, I take it.'
Cupid's Revenge, iv,3.
Sometimes it is beard for the washing..."
Thanks to 1006a for the additional references:
"It looks like the "head" version is quite a bit older; it is listed in the 1660 Howell's Dictionary under "English Proverbs" (unfortunately without a gloss), and again in James Kelly's 1818 A Complete Collection of Scottish Proverbs, where it is grouped with the saying "He will not give his bone to the dog" and both are said to be "spoken of sturdy people, who will not readily part with their interest, or be bullied out of it."