Idea

Haven't done this, but maybe you could dye it like a washing machine; get the fibers wet, put them into a rotating drum, and add the dye at relatively high rotation speeds. Drain out the watery dye, and keep air spinning for a while until dry. It might prevent clumping while dyeing.

Idea #2

Haven't done this myself, but what I've seen people do in situations like this is:

1. Remove as much of the liquid as possible before putting the materials down for drying. Less liquid on the outside means less stickiness and thus less clumping ability during drying
2. Spread out the materials as much as possible over a larger area (baking sheets or wax paper etc) to minimize contact between materials
3. During the initial drying time move around the material frequently enough to break up any large clumps before the material is fully dry.
4. In the dry product you might have to mechanically break up any remaining clumps of material by means of coffee bean grinders or blenders.

Similar approach

Look at Joe Fugate's method of dying homemade ballast.  That might work with dye or thinned paint.

Might try RIT DYE

for dyeing cloth.  The question is are the fibers porous enough to accept the dye?  Rig up some sort of a screen container to hold the wet fibers and dry them with a hair dryer.  Maybe two food strainers back to back.