What a cute project. I love the primitive embroidery patterns. I like the blue water soluble markers like Dritz Mark Be Gone or the Clover one (Walmart carries it). You can also use Crayola ultra clean washable markers in the fine point. I have really good luck with those and they wash out even if you iron over them.
https://www.walmart.com/ip/Crayola-U...Count/19310944
Like any marking tool, it is always a good idea to test if you can. Not sure if you have enough fabric in the kit though.