I don't doubt your knowledge of that "sport" for a moment, but the available evidence suggests that "bar" is a mistake for "car". That may well have resulted from cock-fighting terminology, but you'd need to point to an instance of "bar" preceding "car" (and explain away Napoleon's chariot, mentioned earlier, and the reference to Mars) before you convince me.

I'm not "supposing" here. Have a look at the contemporary and near-contemporary broadsides, as I suggested. You'll find a fair selection at  Bodleian Library Broadside Ballads.

Most begin "The ancient sons of glory", but there are others that don't. Of those containing the "cock" verse, I think that one only has "bar" rather than "car"; and that a relatively late example. Do look into it. If I turn out to be wrong, we'll all have learned something useful. If, on the other hand, it turns out to be you who was "supposing", we will at least have eliminated a red herring.