I've never pulled a sickie and never would. I think it's wrong, and even if I didn't, I'd be superstitious about pretending to be sick when I wasn't.
Be honest about it - you are lying to your employer and stealing a day's worth of pay. You may hate them, think they deserve it, and want to do that but as I say - at least be honest that you're skiving rather than pretending it's 'essential for mental health reasons'.
People who aren't well enough to work - for either physical or mental health reasons - don't have to ask/weigh up whether to work or not. In the case of mental health, they are crippled with anxiety or depression and literally unable to work. They are not considering going to their GP who 'won't be able to tell the difference' and sign them off even though they're basically fine. They are ill, just like physically ill people are ill.
You're not ill, you're knackered - which I have massive sympathy for. I was knackered at work when I had two young kids, and I was lucky enough to have a husband doing 50%. But none of this is your employer's problem - you have a contract with them, and just as you'd be furious if they decided not to pay you for a day you had worked, they have the right to expect you work for the days you're paid, unless you're actually sick. Can you not ask a friend to help you out for the day and babysit during the holidays? I'd totally do this for a single parent friend who needed help.