The rap duo Bob Vylan (named for Robert Zimmerman of Hibbing, Minnesota).
That’s not all: the guitarist’s stage name is “Bobby Vylan” and the drummer goes by “Bobbie Vylan.”
Not a band, but British singer Gerry Dorsey took the name of a 19th-century German composer and became famous as Englebert Humperdinck.
Oddly enough, Kurt Vile and the Violators is not an example - the lead singer’s name (since birth) is actually Kurt Vile.
What about “The Dead Kennedys”?
Calling yourself Uriah Heep sounds like something the Uriah Heep character would do.
Beat me to it, including the line from “Have a Cigar.”
Crazy Horse
Maryann Cotton has no one by that name in the band, but is named for a 19th century murderer.
“Pearl Jam” started as “Mookie Blaylock”, a pro basketball player, but changed their name prior to their first album.
OP says it has to be first and last name, and that’s two first names.
Bastard Sons Of Johnny Cash
Ok, I have an obscure one, the German professional cover band Die Juhnkes. They were named after Harald Juhnke, a legendary, beloved Berliner actor, comedian, entertainer and singer, who was also probably the best known notorious drunk in Germany, and very publicly so. So the band name was tongue in cheek and promised good fun OTOH. Their lead singer is a good friend of mine, we went to school together. They are most probably the band I saw most often on stage. They disbanded when their guitarist tragically died from bone cancer.
Pink Floyd is a fictional person. So despite being named after the first names of two real blues musicians, nothing says that Floyd can’t be the surname of a fictional person.
The greedy music exec in “Have a Cigar” assumed that “Pink” was the first name of one of the band members, and that presumably “Floyd” was their surname.
Even though Pink Floyd is my all time favorite group, they would not meet the criteria I considered when writing the OP.
With your criteria I don’t think there will be many with the full name. If you count partial names you have bands like Tesla and Creedance Clearwater Revial.
Brian Jonestown Massacre
Paul Revere Dick dropped his last name when he was young, so the band is actually named after him. I don’t know if he legally changed it, but his tombstone is inscribed simply Paul Revere.
The point of which was to demonstrate that the exec really had no idea about the band.
Pink became an actual character for the purposes of The Wall, but the name came about because when they were going by The Tea Set, another band with the same name slowed up to a gig, so Syd Barrett thought of a couple of jazz musicians (Pink Anderson and Floyd Council) he had recently seen on liner notes and put them together (at least that’s the story). It wasn’t intended as a fake character, it was just expedient.