First you have to identify the weeds before knowing what if anything to apply. Rather than allowing them to grow so tall either mow or weedeat them before they get that tall and have seeds.

You also do not want to be near any weed killing chemicals until the baby is here because of posible damage to the baby. So identify, mow until after baby arrives, then see what is needed. I would be very worried about the baby as it gets older being injured by falling on the rocks.

There is nothing that will kill "weeds" for good because plants produce seeds that land on areas where those seeds germinate and grow and become the "weeds" we all love to hate. Your current means of control is limited to mowing to aid in keeping those unwanted plants from producing the seeds that will become more of those plants.
As you have found those very expensive "weed" killers do not and thye have the potential of not only greatly harming your environment but you and your baby.

Thanks-that's a good idea. The baby won't be around the rocks, next summer we'll either pour a patio or lay pavers on one side and keep her off the other side. Eventually we'll get rid of all the rocks there too, tarp it and cover it in mulch for a play area.

Lay a giant tarp down to kill the weeds off. This works guaranteed.