score:4
After a long while it seems I'll have to do my best to answer my own question.
On a whim I contacted no less than Prof Mary Beard herself. She wrote a book on the Roman Triumph in 2007, which is extensively quoted in Wikipedia. It goes into details on the religious-political meanings of the triumphs and their format.
It also describes that with the arrival of the Principate and then Dominate, the number of triumphs declined radically. Only the emperors were allowed to have them anymore - even if the victory was achieved by some general or legate.
Asking the question of Prof Beard, she kindly but briefly replied:
The truth is that we assume it was standard. But the only detailed evidence we have is re Caesar. I agree it wd be great to know what they sang about the others. M
To my clarifying question concering the distinction between the Republican era vs the Principate/Dominate, she once more replied
I guess so. It may have happened in imperial triumphs too. I donβt think there is evidence either way. M
Barring new information popping up that is likely the state of knowledge as it stands.