Nick here from Shopify.
Great question. There are a lot of Print on Demand companies in the Shopify app store, which I'm guessing you have investigated already from your message. I have linked them all above in case there was any you might have missed or if you want to reach out to them directly and ask if it's something they don't do, and if they don't if they would consider doing something like this in the future.
I did find one which could be worth looking into called Print Aura. I'm not 100% if they do this. I took a look at their FAQ page and it nothing stood out, but it could be worth reaching out to them directly here to be sure.
All the best,