I found a way to play ps1 backups on a v12 slim model without any mods or even a swap magic disc.
1) There are 3 methods to prevent swapping in the new PS2. I'll assume you know them and defeated them to make swapping easier (if not, and its allowed, I'll elaborate)
2) Boot a real PS1 disc (I chose FF7 Disc 1)
3) At the PS1 Bios logo, swap with your backup (any game doesnt have to be backup of same disc)
4) Enjoy playing your backup.
Simple as that.
Note: Instead of #3, You can also swap in the browser once the PS1 Disc icon appears. But with the method of swap-preventing bypassing i have, its just easier to do #3.
Edit: This also works to boot homebrew apps such as "It might be NES" NES emu. Also, I apologize if this method has been already discovered, I searched the forums and didn't find anything related to this.