If you upgrade from PS3 to PS4, the only thing that happens is that you are unable to play the PS3 version using the account that was upgraded. You are still able to access the downloaded version of FFXIV on the PS3 using another account/user on the PS3, provided they have a PSN Account linked with an SE account and it is currently active.

The only thing that would need doing is for another PS3 version of the game to be purchased by your brother (digital or disc, it doesn't matter as both provide a download voucher code). Use the code on his PSN Account, and then start the FFXIV client. It'll ask if the account is linked or not and if you wish to make an SE Account, provided it is not linked, which will then be linked to the PSN Account in-use.

You do not need to download the client using the registered code on his PSN account. However, you still need to register it as it is proof of purchase and that's what enables you to play at all with said account.