Add special item to each product in cart in Woocommerce

I am trying to add an option to each product in my cart to add gift wrapping to that item, like in this image. Ideally when the user adds it a popup would appear with several options to choose from, and when chosen just close back into the cart page with that gift wrap selection.

Any idea how to accomplish this via code or plugins? (Paid is fine.) Cross sells appear under the cart table and I don’t mind moving code around to simply have the title appear under each product title, but the popup functionality is what I really want as well as having the gift wrap selection appear underneath each product, not as another product.