I've been sensible with closing doors etc but the cat now seems to think it's a game
Your post really made smile (sorry - I know it wasnt meant to be humorous). You've summed it up here, "your" kitten does think this is a wonderful game. A cat will often make a home where it feels safe, under and in your bed clearly being two locations where she feels safe.
I agree with previous posters regarding the water, you dont need to pay for any expensive water gadget, a little water pistol (like a young child would have) or we have a spray bottle (that was originally used with ironing). All you need to do is spray her rump/back (no need to aim for face!)
Alongside this though, and more importantly to ensure its effective, you need to go back to the owners. They got a kitten because they want a kitten, offering to come over and pick her up is not enough, they need to start creating an environment where the cat feels "safe".
They should encourage the cat to "scent" cushions/rugs (rub them up to her gently).
They should feed her and associate a noise to the feeding (like shaking a tin if its dried food). They should try and find a favourite food/treat and do this.
They should close their doors and keep her in for a while so she starts associating their place as home.
Finally how old is the kitten? Sometimes kittens are given away a little bit too early which acerbates these kind of problems. If the kitten is very young the owners should try and place an alarm clock (that ticks) under a blanket near where she sleeps which is meant to replicate the mother's heart beat and encourages the "Im safe" feeling. Needless to say please make sure the alarm clock is not set!
Good luck.
cas.