Hair dye isn't actually that bad of an idea. Except that it's proven to wash out over time.
For this winter's attempt on the wheel well liners, I tried a new approach. Instead of covering the felt material, I tried to transform and protect.
I applied a solid 2x coats of black fabric paint (spray can for outdoor furniture fabrics) and then "sealed" that with a weather protectant spray. I had some extra MudSlinger sitting around, so used that. The protectant spray went on in the fall and I applied another coat after a big wash back in January. We are starting the meltdown period here in Maine with days of mixed snow/rain; hence, holding off on the next big exterior cleaning for a couple weeks at least.
From my experience, removing the wheel wells may seem like a pain, but in order to work with them easily, I suggest doing so. Plus, removing the wells gives you easy access to several other suspension components, if you want to poke around, clean up, etc...