It's an awkward sentence. "Among themselves" makes more sense with a plural verb, but "The crowd have been fighting" sounds odd in AE, no matter what follows. Most AE writers would probably phrase the whole thing differently.
There hasn't been any BE input in this thread so far, so I think it should be said that in BE there's nothing wrong with:
"The crowd have been fighting amongst themselves since morning."
Perhaps another speaker of BE could confirm that I'm not out on a limb here.
To me it sounds very strange with a singular verb. I read it as "The people in the crowd have been fighting..."
But "The crowd has been fighting against some other force, outside itself".
The British are quite liberal using 'have' in these cases. I hear it on the telly.
They even say plurals where none are logically needed as "The team have won."
The American do not use the plural as much, though logically it seems to fit the OP example.
Yet nonetheless "The crowd have been fighting among themselves" sound odd to my AE ear.
I'd say, "Members of the crowd have been fighting amongst themselves." or "There has been
in-fighting among the crowd" "There have been in-fights within the crowd."
I do not know if this is a BE/AE thing, but for me a crowd of people can fight among themselves, in which case the crowd is a collective of many individuals and the verb is plural (are).
In fact, I completely agree with Velisarius, post 11, now that I have read it carefully. It is like I wrote it, really