Money belt through security. It depends on the type of machine used at airports you are transiting. Assuming you are not putting metallic objects in the money belt, you can wear it through traditional metal detectors. If you have to go through a full body scanner, you have to removed it as it would shows up in the animated image. What happens afterwards depends on the airport. Some airport just hand metal detector scan or pat the area. Others may yell and demand you to remove it and make you go through the machine. I usually go through two airports both using full body scanners, so I use a neck pouch instead. It is easier to remove a neck pouch and put in my zippered jacket pocket at the machine than a money belt.

At some airports, there are cash sniffing dogs. You might be tagged as someone carrying a lot of cash. The officer would probably ask you simple rhetorical questions. They are probably not looking for simple tourists needing cash for apartment payments but carriers of serious amount of cash from questionable sources.

Make sure one of you goes through security before the other, so that person can watch for the money belt when it goes through. I would be extremely surprised if you didn't have to remove it.
The foreign exchange rates in US banks are usually pretty bad, but I would still change the cash you need on arrival before hand. Check the bank rate with that from AAA and American Express if that is convenient for you.
Remember to check not just the rate, but also the commission charges, as some places offer what looks like an attractive rate and then charge you another way!
ATM cards are great as long as nothing goes wrong. Personally, I don't rely on my ATM card more than I have to. It is good to have some cash on hand for emergencies. I have experienced broken machines, empty machines, and my worst ATM experience when my card didn't come out!!!