Most carpets nowdays are of course synthetic. So, your burn area is really a melt down.
In my opinion, there is not much that you can do but to camoflage the area.
1) use a throw runner rug for infront of the fireplace to catch other sparks. These can easily be replaced after many sparks hit.
2) Cut out a perfect square where the burn is located and insert a new patch in that area and use carpet glue to secure it down. The seams will show to an extent, but the burn area will be gone.
Use this procedure like a hole in wallpaper or sheet vinyl flooring.
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