If you only want customers send a booking request to authors instead of redirecting to checkout page to complete booking, you can use General Inquiry element. The inquiry form element has same options as General Booking element to configure inquiry forms as booking forms ( check documentations above for more details )
And this is how it will display on single listing page
The customer and author can manage inquiry posts from front-end dashboard -> Bookings menu screen.
You also need the General Inquiry element form to enable booking via WooCommerce. Check this documentation for more details WooCommerce Booking