NDR's satirical show extra3 broadcast the song "Erdowie, Erdowo, Erdogan" on March 17th.
The song mocks Erdogan's thousand-room presidential palace, famously built over a nature reserve bequeathed by modern Turkey's founding father Mustafa Kemal Ataturk.
It further wounds the Turkish President's pride with footage of him speaking in a high voice.
"A journalist who writes something that doesn't please Erdogan will already be in prison tomorrow," the singer intones.
"The time is ripe for his great Ottoman Empire."
Further footage in the video shows Erdogan shaking hands with German Chancellor Angela Merkel.
"Be charming, because he holds you in his hand," the lyrics continue.
The story became public late on Monday. Ambassador Martin Erdmann had been called into the Turkish Foreign Ministry on March 22nd to justify the broadcast by the publicly-funded broadcaster, Spiegel Online reported.
There was no comment from the German Foreign Ministry on the diplomatic spat on Tuesday morning, although Spiegel reported that Erdmann may be summoned to meet his Turkish opposite numbers again in the course of the day.
But formally summoning an ambassador for a telling-off is one of the harshest ways countries can express their disapproval to one another without expelling diplomats altogether.
DPA learned from Turkish government sources later on Tuesday that Ankara had demanded the film never be broadcast again.
Sensitive to criticism
Erdogan has come in for sharp criticism from human rights defenders and journalists in Germany after the EU and Turkey struck a deal that would see refugees shipped back to Turkey upon arrival in Europe.
Turkey does not comply with all of the international agreements on refugees required to make it a valid "safe country" to send asylum seekers back, critics say.
The Turkish President has also overseen repression against media critical of his governing AKP party, with a major opposition newspaper closed by police in recent weeks.
"Erdogan whistles at democracy," another line from the song goes.
"Turkey's ruler Erdogan has obviously lost his grip on the road," German Journalists' Union (DJV) chairman Frank Überall said.
"We've known for a long time that press freedom in Turkey isn't doing so well," the DJV said in a statement on Tuesday.
German journalists have been smarting especially at Turkey's treatment of the press after weekly magazine Der Spiegel announced this month that it had been forced to withdraw its Istanbul correspondent.
"Journalists are being arrested, there are absurd court trials against media representatives, and newsrooms are being placed under state control," the DJV statement read.
The DJV notes that the famous Streisand Effect - by which attempting to censor information only serves to spread it further - is likely to hurt Erdogan in this case.
"Perhaps we just don't understand Turkish humour exactly enough," they joke, suggesting that Erdogan actually wants people to laugh at the video.
Meanwhile, Norddeutsche Rundfunk (NDR) editor-in-chief Andreas Cichowicz told DPA that "it's not compatible with our understanding of press freedom and freedom of opinion" for Turkey to demand the film be suppressed.
Satire show loves attention
extra3's Twitter account appeared to be revelling in the extra attention on Tuesday, posting an image naming Erdogan "employee of the month" and linking back to an older song mocking Erdogan as the "Turkish Mickey Mouse".
Meanwhile, ordinary social media users were quick to contribute their own salvoes against the Turkish president.
"Hey extra3, you need versions in English and Turkish so Erdogan can go viral worldwide," one person wrote.
"Instead of calling someone a killjoy now I'll call them an Erdogan!" another wrote.