In theory, sure. But you'd need to convince the owners to sell. Rochester is community-owned, and they likely wouldn't sell because of fears the Nats would move the team (Syracuse had similar concerns, which is why it is likely the Mets were able to buy them but the Nats might not have had the same chance).
Buffalo is owned by the Rich family, which used to own the Bills. Their stadium lease is up this year, so it's a theoretical possibility, but there doesn't seem to be any chatter that the team is for sale. Charlotte is also owned by an individual, and they have the best attendance of any minor league team anywhere and a long lease in a nearly-new downtown stadium; it's unlikely they'd be for sale at any price the Nats would consider.
I'm not sure you'd get far enough for the antitrust exemption to play into things at all. The problem for that would be assigning players to the independent league teams - the league agreements for those leagues don't allow players to be on contracts affiliated with a major league organization. Nothing would prevent them from buying such a club, however.
There was reportedly another team making a play for Syracuse along with the Mets, though my internet sleuthing hasn't been strong enough to uncover said team. Not sure if that would've been the Nationals, as there's little precedent for minor league team ownership there. Rangers make some sense if they were aware of their eventual split with Round Rock; they own most of their teams.
But like you said, the Mets were seen as less likely to move the team (and have since signed a long-term lease).
