I know this isn't what you want to hear, but Etsy won't make an exception for you. If any seller on Etsy forgets to update tracking, puts in the wrong tracking number, or fails to notice that a connected service doesn't transfer the information (i.e. using a 3rd party label service like Shipstation or PirateShip) it is considered the seller's oversight/mistake. Etsy does not make exceptions.
You need to check your Etsy shop regularly to make sure everything is copasetic.
@RambleOnDesignsCo: Unfortunately @bradgoodell is absolutely correct. Integrations can fail. It is the ultimately the shop's responsibility to make sure that not only are orders shipped on time but also that they marked as completed / shipped on or before the scheduled ship by date. It does not matter if you do this manually, or use an integration be it Printify or Etsy Labels or anything else.
"... manually input the tracking information to get rid of the "overdue shipping" notification ..." Probably good advice. The order will still be considered as shipped late (because you cannot back date the completed / shipped date) but that is preferable to the system thinking that the order has not shipped at all.
Long before Star Seller was even thought of, we got into the habit of spending a few minutes daily checking the previous day's shipments to make sure that they were at least "in transit" so if something was wrong we could fix it within the 72 hour window we are allotted.
As an aside, you should be more concerned about using other people's intellectual property (IP) without proper attribution. You need to include the licensing information for any trademarked and copyrighted names and images in the actual listing, otherwise buyers (and Etsy) may conclude they are counterfeits. Including the licensing information will show that you are not selling those items illegally.
I have also been having a similar issue; Etsy isn't recognising the tracking codes provided by Printify. On Printify's end, my orders say an original tracking code and then a newer one. I don't know why this is! I also lost out on my Star Seller badge because of this, but had been manually adding a tracking code to my orders (only an issue with the international ones) and that was restoring the tracking to 100%. But one order hasn't got the latest information so it's very frustrating!
I have been emailing with Printify about this! Their technical team is looking into it...!