You cannot do this via script. You will have to either use a third party shipping rate application or create or own, to modify shipping rates at checkout.
Additionally, you can also considering using a shipping customization app to show a particular delivery rate only when a particular date is selected.
P.S. - I am the founder of ShipMagic app and both of the above functionalities are possible using our app. But, you would need to somehow send the data being selected in the Buunto app to our app. This can be done using a script to set the line item property.