Adding functionality to the Cart page on BigCommerce

General information for the website: We sell food items online for delivery (takeout)
Kind of development: Customization of existing website
Num. of web pages/modules: 1
Description of every page/module: The "cart" should give the customer an option of when they would like delivery
Description of requirements/features: I am looking to achieve the following functionality for a food delivery website built in BigCommerce. The business delivery breakfast on saturday's and sunday's between 9am and 2pm.

When reviewing their cart, but before clicking 'proceed to checkout', I would like the customer to have two choices:

1) delivery within an hour of placing order, OR
2) delivery on a certain future date (saturday's and sunday's only), within 30 minute windows (9am to 2pm)

I then need this information adding to the order details which are passed through to me when a customer completes an order

I would also like it to be easy for me to temporarily grey out one option from time to time, ie so customer's can't choose "within an hour of placing order" option on a Tuesday, for example, when we are not open for business.

I also need this to work on mobile.
Extra notes: