If I'm singing something American, I tend to take a more conversational tone, and if it's from overseas, I sing it "straighter," although I wouldn't use a bel canto tone for either, since that never sounds right. In general, if I don't sound like "me" singing whatever it is, I don't want to sing it publicly until I do, because it will feel fake. Also, the perception if you change your accent for the song seems to be that you don't have your own voice, although certain people can get away with it for some reason (Ramblin' Jack can get away with a Scots accent in "I Belong to Glasgow," and I have no idea how he makes it work, but it does. Besides that, he gets away with a country accent every day, although it could be argued that he's been doing it long enough that it's become his natural accent).