Thanks for that. It helped a lot!
The badge would not show up with the Special Offer Discount I created. I had to go to your site and see exactly what your Special Offer was. Once I set up the BOGO for a product on my dev site the same as you did the Special Offer Badge showed up.
In skin/ideal_responsive/customer/main/cart.tpl
Around Line # 48 - Comment out the following code.
Code:
{if $active_modules.Special_Offers ne "" and $product.have_offers}
{include file="modules/Special_Offers/customer/product_offer_thumb.tpl"}
{/if}
This removed the badge from my dev site and hopefully works for you.