Of the 2 bikes, I also recommend the Bianchi.
I've built 2 touring bikes from old MTB frames. I want to challenge the received wisdom that it is just as expensive as buying a new bike. I think that refers to adding the same new components that a new bike would have. If you're willing to take your time and compromise where necessary, you can do it a lot cheaper. If you can spend months scouring eBay for cheap parts, you can get good stuff. Of course, time might not be on your side here.
BikePedia says that that Bianchi frame is a mix of chromoly and hi-ten steel. Cromoly is great, hi-ten is heavier. Shouldn't be a problem, but be aware that you can get lighter, and the Nashbar (aluminium) would be lighter still. If the bike is stock, then the derailleurs are Tourney. That's the lowest quality Shimano MTB level. Personally, I'd be a bit nervous about them. It's shame you couldn't get one of those Trek 820s. They probably have better components.
The Burley Nomad trailer seems really expensive. That would be most of your bike money gone in one hit. I think I can see a braze-on in the photo, in which case you wouldn't need special mounting p-clamps for a rack as Valygrl suggests. If you can travel light enough to not need a front rack and bags and just use a rear rack, that'll save you some cash.
It looks to me like the chain on the Bianchi is either broken or sagging. For me, what you might need to do first is get that bike in decent running order, and find out if you can ride it long distances with the current setup without wrist pain, neck pain, etc. From there, it would dictate your next move. The Bianchi currently has grip shifters, which I'm not sure you can put on a butterfly bar. Anyone?