You have several options at your disposal. The first and likely most straightforward is to simply enrich your line items with additional fields that point to these promotions and then edit the View to show those fields. It appears you've already edited the View to remove things like pricing, so it should be easy enough for you to add in an additional custom field that you've populated on the Add to Cart.

Another option would be a custom line item title callback. Using hook_commerce_line_item_type_info_alter(), you can specify an alternate line item title callback that is capable of displaying the offer along with the product title. The one drawback of this is that the promotion would be included in links that use the line item title to connect the item in the cart to its display page.

Finally, you might consider a custom Views field. To render the field, you could examine the contents of the $line_item->data array (or perhaps its price components) to find what offer applies and render it accordingly. This is probably the most future proof but also requires the most coding; I just like that you won't have to manage static storage of the promotion type in some field on the line item like you would if you go with the first option.