Hi Almond,
That sounds like a fun idea! I know turmeric is sold commercially for cats and dogs as a supplement, with curcumin content starring as the active ingredient. Turmeric is great for adding color to food so I don't imagine much is needed for white fur. In terms of textiles, it is a rather temporary dye that fades and washes out easily. That might be a good avenue to try but I also wasn't able to find any authoritative sources stating explicit toxicity or safety, so you will want to do some research first.
In another vein, the ASPCA maintains a fairly long list of toxic and non-toxic plants for cats. Along with the trusty control F function, it can be quite valuable if you are looking to check dye plants you already know about.
https://www.aspca.org/pet-care/animal-poison-control/cats-plant-list
Hope this helps!
Eric S.