How should I distribute the weight on a marble island?
Even without knowing how thick it is or how heavy, that's a lot of marble.  You really need to attach this to something, somewhere.  The wall, the floor...  Without that, you would have to have a serious counterweight at the bottom, at least as heavy or heavier than the marble.

Why not just attach it to the wall or the floor?

When your child's health is at risk, I would double-up on precautions rather than find the bare minimum level of acceptable safety.

Your three-year-old will be stronger than you expect, and even if not actively pushing against it, could fall against it, setting off a chain reaction.

Think for example, if you have something on top of the island, like a vase.  Even if you manage to keep it from tipping while on wheels, it can still move.  If your child pushes or falls against the island, the island can move in one direction, the objects on top will move in the opposite direction and tend to fall on the side that your child is on.

I would not risk this situation with my child or anyone else's child, even if the top of the island was made of wood.  Whenever it falls over, it will hit whatever is on the otherside like a sledgehammer.