Many bioluminescent creatures on Earth are semi- (or very-) transparent with the illuminating component inside the body where the chemistry is protected — where it can't be corrupted or washed off.
If you want something akin to "reality," consider making the fur semi-transparent and with the bioluminescent inside the fur for the same purpose.
It's somewhat unbelievable that a bioluminescent creature would glow noticeably in full daylight. But it is believable that the glow would create an under-color to the grayish, shimmer caused by the semi-transparent fur. Personally, I think this would not only look cool, but it would give the creature something a a chameleonesque camouflage during the day as the reflection of surrounding colors would combine with the creature.
At night the creature would look blurry due to the semi-transparent fur. Almost like a big wil-o-the-wisp.
Finally, it removes one aspect of the creature you might have been depending on: the fur, once shaved from the body of the creature, would stop glowing. This is because the hair is removed from the follicle (and the chemistry of the body). If you were counting on harvesting the glowing fur as an aspect of your book, this would be unfortunate. But it would make the creature very hard to prove exists since no glowing fur would ever make it back to the village.
IMO, you can believably have an all-body bioluminescent creature using the same mechanisms terrestrial bioluminescent use, just on a somewhat larger scale. Evolutionarily you'd have trouble justifying the creature, but do we care?