No, it's not cheaper. You have to factor in new driveshafts, new crossmember, shifters, and any other rerouting along with the doubler cost and it EASILY surpasses changing gears. The extra length, unless you run a Titan shorty version, usually causes rear driveline issues in short wheelbase vehicles and quickly turns into a high dollar CV type rear setup along with a potential diff rotation to go with it. There's much more to a doubler than just the cost of the gearbox itself.
I do really like the Black Box and plan on running one in my setup (I've been spoiled with 203/205 doublers in the past), but there is a lot to consider when doing so. And as @bigwalton said, you have to consider your end goal in developing the combination of ALL parts that get you where you need to be in terms of capability and drive-ability. I'm mainly looking at it so I can keep the axle gears relatively quick to sync with the OD in the trans and maintain decent road RPM's. I'm also planning on a manual, and I believe manuals benefit far more from additional gear reduction in true crawl type situations. However, I'm far from that point in my build and a more compact Atlas does look tempting.
Another thing to think about, especially with a manual, is how many unique gear options are you actually gaining? Assuming you are running the AX-15, Black box, and D300 you only gain 2 gears; 1st-low-low and 2nd-low-low. The Black Box and D300 are almost the same, 2.72 vs 2.62, so there is no gain from one low range to the other low range. In double low, 3rd gear is now the same as 1st in single low (10.3 crawl vs 10.4 crawl), and 4th is very close to 2nd in single low (7.1 crawl vs 6.3 crawl). So, all that work adding the doubler and all you gain are 1st-low-low (27.3 crawl) and 2nd-low-low (16.6 crawl) as unique options, and have still done nothing to address drive-ability of your rig's configuration as a whole.