Hi Angel. My girlfriend had a similar issue and we suffered through about 4 hours of headache before finding a solution, so give it a try:

1) Log on to your PS4. Close out FFXIV if you have it open.
2) Go to the playstation store.
3) Scroll all the way down to the bottom and click Redeem Code.
4) Put your 12 digit early access code in here and it should prompt you with a 1.7 GB Heavensward Download.

This is the patch you are missing that is preventing the game from recognizing you have early access. Once the download is done, restart your PS4 and try again.

Be sure to do this FROM YOUR PS4, NOT THE PSN WEBSITE. The reason my girlfriend's copy wasn't working is that we put in the code on the PSN website, which flagged her square enix account to add early access, but didn't flag her PS4 to give her access or actually download the patch (WTF....). We tried the solution in this thread and it only prompted us to buy the game instead of letting us download it.

If this fixes the issue for you, please help us out by telling everyone else you see having this error about it.