Clever idea, but my immediate reaction is that chickens have very different digestive systems, so the chemicals involved will be very different. I'd be genuinely surprised if you get stones from a chicken.
Incidentally, I'm not sure your objection to the test is all that serious. If the stain doesn't enter the center part of the stone, there has to be some reason for that. The only reason I can think of is that the center of the 'stone' was not penetrated by the coloring. The only way it could escape being colored is to have started out somewhere were the coloring wasn't present--that is, in the liver or gallbladder.
Of course, it's conceivable that the stain might penetrate a genuine gallstone, but that seems unlikely to me, since these stains aren't that aggressive. If you stick most solid non-granular substances in a dye,
only the parts in direct contact with the dye will be affected
Chlorine gas is colorless. Gases aren't usually stained by dyes, so that's not a real concern here.