According to an entry in Bob Brownell's "Gunsmith Kinks", sawdust from wood of the desired color can be mixed in as a colorant. I seem to recall having tried it years ago and, if my memory can be trusted, it worked pretty well. Real fine sandpaper, on the order of 320 or 400 grit, would be called for. Like the dye, it doesn't take a lot.
Could always mix up a small amount of epoxy and add a drop of brown paint to test whether the paint is soluble in the epoxy and if it darkens it enough. I've mixed small amounts of Acraglas many times by counting the drops from a toothpick.