I think I can help, @Anna595. Your place is unique, so people have other ways - via friends, newspaper story, etc. - of hearing about it and getting in touch to book? They are not internet savvy, or they are reluctant to sign up with Airbnb and do the verifications? And you would like everything to go through Airbnb. I am often in the same boat.
There is no way to just set it up for them through the site, in the way of a generic receipt, and then say 'here is what you pay'. Airbnb doesn't work that way. They have to set up their own accounts, and they have to initiate the booking on their end. You cannot do it for them.
When I get calls from people wanting to book, I tell them it has to be through Airbnb. They give the excuses with which you will be familiar. I say how sorry I am about that, but it is all I can do. They can get quite pushy. They tell me I am being unreasonable when all they want, etc. Stick to your guns. If they want to come badly enough, they will figure it out.
Is this what you were asking?