Re: BF 4 Premium not working on PS5.
Its weird if it is the same account you use on both consoles, that not everything is available which you actually have on PS4 already.
But i got another idea what you can try.
You can remove the game and all its add-ons again on the PS5 and then use the PS4 --> PS5 data transfer option to transfer everything BF4 related you have on the PS4 atm. (i guess on PS4 you got everything unlocked and installed)
check this guide: https://youtu.be/8pWB4YicSug
If you haven't tried it yet, maybe this solves it. 🙂
Greets,
Travis