The phrase "Bart-mangled banner" was used in an earlier episode in which Bart ruins Lisa's school project: a new state flag for...whatever state Springfield is in.
In jail the family are shown a piece of propaganda film which contrasts the US Constitution with the Bill of Rights. The Bill of Rights is technically part of the constitution (they are the 1st ten amendments), it was made only a few years after the original Constitution.
In addition to giving all of his eight children, patriotic American names, Apu also mentions that he has changed his family name from Nahasapeemapetilon to McGillicuddy. That was the maiden name of the character Lucille Ball played in "I Love Lucy," surely the most all-American sitcom of all time (despite Desi Arnez being Cuban). It is obviously why Apu could not have used Lucy's married name, since it was Ricardo, which was no more "American" than Nahasapeemapetilon.
The title refers to the American national anthem, "The Star Spangled Banner", named after the flag flown over Fort McHenry when the British bombarded Baltimore MD during the War of 1812.
(ACLU) "The American Civil Liberties Union" is a nonprofit organization whose stated mission is "to defend and preserve the individual rights and liberties guaranteed to every person in the country by the Constitution and laws of the United States."