Ok here's how to get into the beta if you're having problems.
Step 1: Get your beta code from the bungie site. That's just perseverance. Keep spamming until you get through.
Step 2: Don't even try redeeming your code through your PS4 you'll end up flipping a table. Instead read
https://support.us.playstation.com/...-to-redeem-a-code-or-playstation-network-card Click "PC or Mac: Redeem Vouchers through the SEN website". Follow the steps to redeem your code.
Step 3: After redeeming your code you will not be able to download the game from the same site. Now you'll have to deal with PSN (god speed). This step was provided by TimeEffect:
Don't use the circle button like I was! Follow this procedure.
Step 4: Go down to the Destiny pg on the PSN store and click it. Keep following the quoted method whenever you get an error screen. If the download button doesn't work just keep trying the quoted method.
Step 5: You're in! Or you should be unless something changes. You can't actually play unless Destiny can connect to PSN which I haven't solved yet but hopefully you'll be lucky and won't need additional assistance.