Even with high ping, application layer is probably the wrong place to solve this problem. We'll likely get email working as one of the first problems and be back to correspondence chess and the like. Even Civilization 5/6 works over email.
This is actually a totally solved problem using a solar reference frame and done every day for missions far out in space. NASA has a free tool[0] to see the current time for any body on earth, even correcting for light-delay. It works via email, ftp, telnet, and web.
But you can not send information. Which is information that you didn’t send. Thus, you sent information.
This is like saying you can’t send information with electricity. You couldn’t at first (FYI), but then someone invented Morse code and the absence and non-absence of electricity made all the difference in the world.
You can send information by sending additional information through classical means (thus limited to relativity). This is the basis for quantum cryptography, where you can have a shared one-time pad that is impossible to intercept without detection or and impossible to copy (thanks to the no-cloning and no-teleportation theorems).
But you can't send information through quantum entanglement itself without rewriting a significant chunk of physics as we know it. If you're going to do that, all bets are off and you should be consulting with wizards and not physicists.
