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.
Your question verges on asking for opinions rather than a technical answer. The functions of either type are identical. Stainless has always been considered the best choice for longevity and durability. The newer composites are very popular, but somewhat suspectable to scratching from hard metal cookware such as cast iron. I have seen some disclosures on various brands that warn of staining from substances such as wine and dyes. They also warn against the use of certain abrasive cleaning products, so more care must be taken with composites. They are however, very durable, less noisy and can add style to a kitchen decor.
As far as installation under a stone countertop, this is typically not a DIY project. Granite fabricators take great pains to prep the mounting surface. The sink is dry fitted and the cutting and polishing the adjoining edges is done precisely to the individual units. They are then epoxied in place, usually with the counter upside down, clamped and allowed to cure. If done on site, install usually would have to be done from underneath,(unless the slab is small and can be flipped over) not the optimal way and very hard to clamp into place if not impossible. Ordering the counter with the hole precut and polished needs to be EXACT, or you will be in a world of hurt when you go to mate the counter with your sink. My preferred counter fabricator only changes $85 for installing a supplied bowl. I think that is a bargain and the bond is warrantied.
Best Answer
Assuming you don't have extensive scratching or damage to your countertops, sometimes a simple acrylic sealer/refinsiher will work great. You will need to clean the tops thoroughly with a non abrasive cleaner and rinse completely. After they are dry, simply apply the acrylic sealer with a lint free smooth cloth evenly over the whole surface. Let this dry for an hour or two and repeat the application. After the last coat is completely dry, buff with a soft cloth. Usually two or three very thin coats and a good buff will restore a nice shine. This should last a year or more depending on the amount of use the tops get. Do not use abrasive cleaners in the future. Simple glass or all purpose cleaner is fine. You may also consider having some anti-bacterial solution (looks like water) in a spray bottle for quick safe cleanups.