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One good natural weed killer involves boiling water. Simply boil water and pour it directly onto weeds. The high temperature will damage the cell structure of the plants, effectively killing them. This method is best suited for areas where you want to prevent any vegetation growth, as it doesn't discriminate between weeds and desired plants, however this is my preferred method.

Here are some more tips:

https://www.gardeningknowhow.com/plant-problems/weeds/safer-alternatives-to-roundup.htm

https://www.gardeningknowhow.com/plant-problems/weeds/using-organic-herbicides.htm

The best way will be to pull what you can, and kill the rest with boiling water. This will leave your mulch and soil intact with no toxic residue. Another thing that you can try is smothering the area with cardboard. I don't prefer this as much, but it is still a safe bet.

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The preferred method for most will be to place cardboard on the weedy areas and wet it. Leave it until everything under it is killed off. Then you can remove and have something else established in its place.

Another option is selective chemical herbicides.
Selective herbicides such as fluazifop-p-butyl, clethodim, and sethoxydim are commonly used to control grasses while sparing broadleaf plants in various settings like lawns and gardens. They selectively target and manage unwanted grassy growth, preserving the desired vegetation.

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It can, but typically a properly turned and heated compost won't contain any seeds. You can use boiling water to kill them off without harming the soil, though.

https://www.gardeningknowhow.com/composting/basics/heating-up-compost-pile.htm

https://www.gardeningknowhow.com/composting/basics/storing-compost.htm

Unfortunately, getting rid of these pests is not easy. Hand pulling will be your best bet using natural methods. It takes persistence. There are selective herbicides that will only target the nut grass, and leave the amaryllis alone. This requires knowing the species. They are also synthetic, however.

Another option will be to dig them all up when they are resting after flowering. This can be a slowdown in growth, or dieback of leaves. Once they are dug up, you can treat the area, or boil water and treat the area daily until nothing grows. Once all weeds are killed off, you can plant again. This is the method that I recommend for true weed control. Continuous boiling water will destroy the weeds, while keeping the soil intact and without contamination.