Are there biodegradable pigments that are soluble in oil, but not in water?
There are several such pigments which are bio-degradable by some microbial action (mostly bacterial action) and are oil soluble (lyophilic) and water insoluble (hydrophobic).

Some are:


chlorophyll : certain bacteria can decompose chlorophyll.
A paper citing zoo-plankton activity on chlorophyll is here.
Xanthophylls : This is a general category of several pigments whose degradation by marine bacteria has been widely  studied. (You can search for many relevant papers online).
Carotenes : Certain carotenes are also reported to be degraded by fungi and mushrooms. For example here


Anthocyanins (and related polyphenols) which have dark red to purple color are mostly water soluble.