Buying BD or BikeIsland, you know what you are getting: a cheap bike. Not a bad bike, but a cheap bike. Likely though, a very usable bike which doesn't need a lot of upgrades to keep on rolling for a while. If you were going to get wheels for the Bianchi, get them for the new bike instead.
Not sure about FSA cranks and BB, I know Tektro brakes don't have a lot of fans but you were going to buy brakes anyway ...just the 105 brifters, cassette, and derailleurs would cost about $300 (I could look it up but won't, too lazy) and they are brand new.
if it were me I might buy the one and build the other because ... hey, it's good clean fun (right now I am finishing an '83 Cannondale, contemplating (and collecting parts for) a dinosaur steel Raleigh frame, and trying to decide if I'd rather buy a couple decent used bikes or go the Chinese carbon route. No, I won't take my medication.)
That Bianchi is one sweet-looking bike. I'd build it for the road because I don't have any gravel roads around here, but I'd bet (so long as you aren't trying serious MTB trails) that bike would be a blast on dirt. Looks like there is tire clearance. (Don't worry about people who say "it might not work on dirt." If your main aim is fun, you will have fun. If you want to race for trophies, you should buy a much more expensive bike.)
Spread it to 130 or 135 and have a ball---and if it breaks while you were out having fun, you would still have the new bike for tours and organized races.
Figure you could do both for $1500, and you could do the Bianchi in bits and pieces--scavenge parts off smaller/trashed bikes and stuff. I'd bet in Colorado you can go to any major city's suburbs the night before trash day and pick up trashed kids' bikes with some decent components, and I;'d bet yard sales out that way are insane.
Since you aren't afraid to wrench, check Craigslist. I bet a state with cycling culture like Colorado's would have tons of used or wrecked bikes with salvageable parts, used bikes you could almost ride on tours or trails right away ... all kinds of stuff.
Either way, plan on spending $600-$800 for the bike, whichever way you go.