One possible solution to this could be to use Shopify's CarrierService API. With this API, you can create a custom shipping method that will calculate rates based on your own rules and algorithms.
In your case, you can use the customer's selected drop location as the destination for the shipping rate calculation. You can then return the rates that you calculate based on the destination provided.
To accomplish this, you would need to create a private app that uses the CarrierService API. You can then set up your custom shipping method and integrate it with your store's checkout process.
To retrieve the location selected on the cart page, you can store the information in a cookie or use the cart notes feature to store the location data. When the rate request is made, you can retrieve this data and use it to calculate the shipping rate.