I think barks, roots and woods in general do give color to CP soap.. There's cinnamon, madder, alkanet, red sandalwood (the real one)... so I'd try one small batch. Maybe as an infusion, or if its fine powder add it to batter.
I've never seen soap with only camwood as a colorant but I know they add it in the African black soaps which range from brown to almost black. They use it in face masks too so I'm assuming it's safe to use in soap at relatively higher amounts (unlike cinnamon).
Hopefully it does give you a shade of red, I'm guessing maybe towards maroonish. I'm looking forward to your tests