Oshikuru: Demon Samurai
According to the credits, the show they were watching was really "Boogiepop Phantom"
This fictional anime production was also mentioned in an episode of "Two and A Half Men" (also a Chuck Lorre production) in which "Jake was helping Charlie write a theme song for 'Oshikuru: Demon Samurai'...." .
The theme song Charlie wrote for Oshikuru in "Two and A Half Men" was shallow and corny, with the lyrics:
Oh Oh Oh, Oshikuru, Oh-Oh-Oh, Oshikuru, My-O-My He's a ... Demon Samurai ... Who's the guy who had to die? Oshikuru!
Jake was familiar with OshiKuru and his dark origins and chastised Charlie for producing a song that didn't reflect the character. He worked with Charlie to write a better version. This resulted in a bonding between the two as they struggled through the exhaustive creative process to produce a work that was truly reflective of the character of Oshikuru, Demon Samurai.
Charlie submitted both theme song versions to the television network, and the network chose Charlie's original corny version.