Drupal Commerce add multiple products to cart at once with different attribute values
Since your not sending a single SKU, you could just overwrite the validate and submit handler for the add to cart form and then, using the existing validate and submit handlers as a blueprint, write your own logic to add multiple products. You'll want to make sure you create line-items for each of the variants your adding to the cart so that pricing is calculated properly, but it should be a fairly straightforward process.