Superb translation!

Still, I have a couple of quibbles.

I'm not 100% convinced of the choice of "bawl." My only encounter with the word "gueule" had been in connection with an animal's mouth or a derogatory way of referring to a person who runs his mouth, putting people down. So I read the article in ATILF about "gueule." There's a ton of uses of the word, a great many of them derogatory. My problem with the word "bawl" is that I almost always encounter it in the sense of crying hard. And I think in this song it would probably mean more shouting, screaming, howling, hollering, I even like the one line where you translated it as "carrying on." I do recognize, however, that bawl can also mean shouting, screaming, etc. So, I guess it's kind of up to taste which way to go.

Ça tient d'la place > I think I would go with "She holds her own."

Since there are none, > I would have gone with, "Even when there are none,"

fait tout un drame ! > I think I would have said, "makes quite a fuss!" or "puts up . . ."

In fact, I think there's some bad punctuation in this passage:
Y a pas, faut qu'elle cherche des raisons
De ça, elle en fait tout un drame
I would correct this as:
Y a pas, faut qu'elle cherche des raisons.
De ça, elle en fait tout un drame !
Which comes out as:
Even when there are none, she has to look for reasons.
And then, she puts up quite a fuss!
And of course, this couple of lines occurs more than once so I would fix all occurrences.

de toute sa hauteur >Your choice of "haughtily" is fine, but I think I might be tempted to continue the idea of her being short here. Something like "from ever last millimeter of her height."

There's a number of spots, too many to list them all, where the French is in the present tense and you translated it in the past tense. I think the present would work as well in English.

I disagree with this punctuation, too:
Moi, ça me fait rire, mais en dedans,
D'abord, je suis un gentleman,
I think it should be this:
Moi, ça me fait rire, mais en dedans.
D'abord, je suis un gentleman,
Which slightly alters the translation:
It makes me laugh, but inside.
I'm a gentleman, first and foremost,
Very subtle, I know, but I think it makes more sense like this.

Je vais lui donner raison ! > This is an idiom. It translates as, "I will prove her right!" Or maybe "I'll say she is right!" That would fit even better with the few lines that follow.

Elle m'arrive là, juste à mon coeur, > I'm not so sure about your choice here. I think this might be an echo of the opening lines: "Next to me, she comes up to here, just to my heart." How symbolic, by the way . . .

Immediately after this line is a whole passage that is present tense in French that you've translated in past tense. Again, I'm not so sure about that. Within this passage, you have translated "des fois" as "this one time." I think it's "sometimes." I take the sense to be that this happens occasionally and the same scene with the hugging occurs each time.

Elle a regard tellement surpris
Qu'on dirait que ses yeux sont punis ! >
She gets such a surprised look
That you'd say her eyes were being punished!

j'ouvre les bras > . . . arms (not eyes)

Pour un peu, > You did this as, "For a moment," I would say "After . . . "

And since I'm stronger than her. . . > I would go with "And that I'm . . . "