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I know that you mentioned it has new batteries but can you try another set of batteries? The reason being is because sometimes and even though it is rare, batteries can come dead even though they haven't been opened yet.
Please let me know after trying a new set of batteries
Did you perform the battery change? You mentioned before the battery change.
It is very likely that the keyboard is having a hardware failure. The good news is that it is under 1 year old and would be covered under a warranty for repair/replacement. The fact that it does power on but quickly powers off is a sign of a power failure somewhere within the keyboard.
Yes, I am still here. It does take 1-2 minutes for a reply since I have to type it out.
It does not matter where you purchase it-- The warranty is a 1 year from purchase regardless of the purchase location.
I am sorry if you disagree with my answer but since it powers off as soon as it turns on is a sign of a power failure. The keyboard is under warranty and can be replaced for you under warranty at an Apple store. I understand that you have viewed different answers on Google but each situation is different.
I understand, Scott.
Since we have tried different batteries and a wireless/blue tooth keyboard is very basic, there are not very many troubleshooting methods available. I understand that it seemed like a quick response then a response to return it but if I suggested more troubleshooting steps, I would be wasting your time when it ultimately should be returned. I hope that you understand that I never want my customers to feel like I neglected them or didn't provide them with a helpful response because my feedback from each question affects my score.