If the option Show to logged users only, which allows you to force users to log in on booking with the booking confirmation request enabled, cannot work, you might combine YITH Booking with YITH Request a Quote plugin (premium version).
So, enable the Quote feature on the booking products and for guest users: then, your users will be able to choose the date and send a Quote request. The admin will be able to check the details and send a quote per email and finally the user can accept and complete the payment.
Please, read more about YITH Request a Quote details here.