If you are not doing the production yourself, you need to use a third party that is approved by Etsy, or get whatever printer you use approved by Etsy. Etsy started off as Handcrafted items and Vintage items. They still make some pretense of that, but there is now a process to outsource your production. (apologies if you are already aware of this)
That said, your plan of listing a shirt for a week and then taking it down is insanity
Internet sales work on search, and search works on SEO, and that means your item damn well better be there at the other end of the Google search results when a would-be customer clicks on it. And people often go "browsing" for gift ideas, bookmarking potential finds and returning to the best ones once they have narrowed down their choices. Will your item still be there when they come back? It costs $0.20 to list an item for 90 days. Use every day you paid for.