Greg, it should not. car fabrics are solution dyed not skien. so the dye is either part of the fiber or inside the fiber. versus an added dye in a dye sight on the exterior of a fiber.
However that being said it is not the best way to approach this issue. You need to extract and loosen any hard dried blood. after that you need to apply the strain HPO no 50/50. when i did this kind of work i used 45% hydrogen peroxide, versus the 3% you get in stores. after the fiber is clean and the per-oxide is on it cover the stains with clean un marked lettered bags. this will keep it from drying out. HPO works in an odd way. Light breaks down HPO this is what causes the bleaching action. IF the HPO dries out before the stain is removed you will have to re wet it. 3% HPO can take up to 24 hours to work, usually wetting every 3-4 hours. (lightly mist) 45% HPO will work in 10-15 min.