Plus, all it really takes in English football for one of those songs or chants to get started is a few fans starting a catchy tune with the words and the rest of the fans join in. For example, using an example for an NFL stadium (though, as you said, it is highly unlikely to happen in the NFL), if we were to play at the Chargers current stadium, an English football supporter might belch out a song in the tune of When the Saints go Marching In such as this, "My Garden shed! Is bigger than this! My Garden Shed is bigger than this! It's got a door, and a window! My Garden Shed is bigger than this!" Then it might shift to, "My budgy's (a bird) cage! Is bigger than this! My budgy's cage is bigger than this! It's got a perch and a mirror! My budgy's cage is bigger than this!" (found those on YouTube). So that's one way you can poke fun at opposing team's smaller stadiums. Could be worth a go for at least a few fans, but yeah, like you said, I don't reckon it'd happen.
Edit: Here's the video: