Pay a dodgy private quack. The BMA may deny their existence, but I know of several.
Alternatively, you could just go to your own doctor and make mutterings about stress at work affecting your well being. Throw in some guff about it affecting your reltaionship and your ability to sleep at night and make love to your partner. Mention that it's making you depressed, but you don't want a chemical crutch. Add a touch of "it's all due to mounting pressure at work" and the Doctor will be signing you off for months.
Remember that workplace stress is the fad mystery illness of the moment amongst the blame culture classes. Most Doctors can't be bothered to argue and will sign you off no bother. It will label you at work forever though, regardless of what equal opportunites bods may say.