On Minecraft PS3 can I set up a game that either of my daughters can access without split screen?

Only with the use of two PlayStation 3 consoles, 2 PlayStation network profiles and through the use of system link.

The only way would be to keep deleting the world from one's PlayStation account and copying it to the other to switch every time, (and vice-versa). Messy!

Otherwise, you'll need to have two PlayStation consoles, run the game on one and play it on the other with a different account (even more messy, and expensive!).

This is a very good and legitimate question, and it's a shame there is not a real, acceptable answer.

The only feasible solution would be playing on a version where running something such as a "server" is possible (either Windows 10 Edition/Pocket Edition or Java Edition/PC), but in your usecase, this is most likely not going to be helpful at all.

When creating a new world that has a local second player, the first player logged in becomes the "server" in split screen and will always be player one. This is also the case if there is only one profile on the PS3 as it will just create a temporary account with the same name.

So no, to answer your question. Only player one can ever play by themselves. Any time the second player wants to play, player one has to be logged in first which will split the screen.

Just go to the Saved Data Utility (PS3) folder and go on the Minecraft world and press triangle then press copy and select the account.

You're going to need two seperate consoles for it to not do split-screen.