I have had good luck with Grandma's Secret Stain Remover:
https://www.amazon.com/Grandmas-Secr...dp/B01KG76NRU/
Rub it in, wait 20-30 minutes, then wash. It does require washing. The stuff comes out of the bottle drop-by-drop, so it will take a bit of pressure to get enough drops for a stain the size you describe. I used this on a pair of white sneakers stained with grease from the bottom of a luggage carrier at the airport. This was after my husband had tried everything under the sun on them, including Dawn and anything else he could find. I used a soft toothbrush to rub in Grandma's, waited half an hour, then washed them in the washing machine. My husband couldn't believe it! He had spent hours on the shoes without results.
Grandma's is safe if you don't know the nature of the stain. Peroxide is iffy because it weakens fabric, can remove color. Dawn and Restoration are safe, as far as I know.
I figure if Grandma's does not remove the stain (or whatever other options you try), applique is your best bet.