Nick here from Shopify.
Great question and I believe there is a way to do this with a workaround within the Shopify Admin/ Shipping Settings. Depending on how you have your current shipping settings set up, you could adjust the settings of the additional $5 charge for "Cold Packing" for products with a specific weight, so that when someone checks out, the weight of the product will match what you have entered in the Shipping settings. The new T-Shirts you are looking to sell, will in turn not be in the same weight class and have the additional charge added to the shipping charge. I did a test for this on my own test store and it did work. Let me know how it goes/ if you have any other questions.
Before I go, Shopify also has a helpful document on how to add weight based manual shipping rates which you can see here.
All the best, Nick