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Yes, there are several grass specific herbicides on the market. Just look for one marked for killing grass. They should be carried at almost all nurseries and hardware stores.

This article has some suggestions that are deemed safe: https://www.gardeningknowhow.com/special/organic/homemade-pet-friendly-weed-killer.htm

I think your best option would be to plant something pretty that would be as tough as the weeds are and will eventually push them out. Creeping juniper (also called vinca minor) is a popular choice. Your only other option, other than spraying with chemical herbicide would be to hand-pull. You would need to pull the jewelweed before it blooms to avoid spreading more seeds.

If you have a few months time, I would actually recommend black or clear plastic. Pin it down over the area and let the sun do the work. It will cook the soil and will not only kill the weeds, but their seeds as well. This is called solarization. This article has more information on it: https://www.gardeningknowhow.com/garden-how-to/projects/how-to-solarize-garden-beds-to-eliminate-garden-pests-in-the-soil.htm

This article should help: https://www.gardeningknowhow.com/special/organic/homemade-pet-friendly-weed-killer.htm

The only weed killer I know of that will not harm flowers and grass is broadleaf weed killer. This will kill many of the types of weeds found in the garden, but not all. It is a chemical weed killer and could be harmful to pets. There are no organic weed killers that are specific to just broadleaf.