Dying Round 1: Combining advice from aranel13 and absinthangel, I used a ton of water and a tiny bit of taupe dye (about 2 teaspoons, actually) on my fabric. Either I should have let it sit in the dye bath longer, or used more of the package, because the fabric is still "Statue of Liberty" green. Not exactly the look I'm going after with this costume...

Dying Round 2: Dumped in the rest of the package of taupe dye, plus a little of the dove grey. Let it sit in the dye bath for a Very Long Time. Ironically, my fabric has faded instead of gotten darker. If I ever make a Statue of Liberty costume, I'm not going to have to buy as much faric...

Final Verdict: Several yards of grey cotton from Joann's. I don't have the time to fiddle with the color any more. And while the fabric I found at some little out-of-the-way quilting shop is a close match to the unretouched reference picture I've been carting around, I just don't feel it's right. More on the green end of the spectrum than the grey.