Hi Lena,
Unfortunately, I don't have any first hand experience with drying a wet bed, but I'd say that leaving it open like You said is probably the best and hustle-free option. Airing Your room might be a good idea as well, but depending on the weather where You live it might be difficult right now. It's really cold outside where I live now, for example 😕
About the challenge though...
First of all - it looks like a fantastic idea! 😃 Especially since from what You said it seems that Your family would be ok with something like that - must be amazing having that kind of freedom! I've also