I believe 'Berk' comes from Berkshire Hunt which is known in cockney rhyming slang for c**t. Thus, shortened to Berk.
Maybe I am wrong and someone might enlighten me :)
Completely unrelated but I love seeing how folks from Berkshire get upset when you pronounce it Berkshire and they spout actually its barkshire nonsense. its not my fault some berk spelt your county wrong years ago and you've never corrected it is it ;-)