Be aware this could be a fairly involved hack and is unlikely to be a simple unlock type code. This is not a simple "more than 10 of these breaks the game so we won't allow it where everything else has 99 or 255" type setup but more of a increase the size of the player party type hack.
Two, or maybe three, main issues to be aware of here.
1) The game might be hardcoded to have the animals (which I presume have their own hunger, happiness and whatever else stats). Assuming you don't care to swap it in and out for effectively another setup (think like having a premade weapons setup for a game that does not allow it in another game) you get to expand the memory allocated to handle it, and such things are often in short supply on consoles, as well as handling any new AI additions (might not apply here but think about what might go if you had the classic every 5 minutes lose one food point, in that case you would then have to expand the food handler to also include your new animal slots, in addition to any other stats or behaviours).
2) As well as conventional memory you also have graphical memory. If the game has to try to render all those models at once (plus any AI) it might be too much for the console.
3) Saves might also be troubled in a similar manner to 1). Being a Wii game you should be able to expand it more easily than a hardcoded limit on an earlier console.
The closest to an easy hack would be find where in the memory the animals are held and create a savestate. Create one savestate with one set of animals, make the save, save the changes to the animals and then swap in the other set, before swapping back later (might also want either a blanket change for hunger/interaction/happiness values for the time gone but cross that bridge as and when). You could possibly automate this to happen when something in the game gets detected (pretend you have a portal to an alternative timeline/dimension in your shed or something) but this would be at emulator level, or possibly want one of those debugging devices, rather than something I would want to attempt in game.