Shakespeare uses some pretty rugged French in his history plays -- "moi" is pronounced to rhyme with "destroy" in Richard II, Act 5, Scene 3 -- but it's not rugged the way Katherine's accidental swears are rugged. In Act 3, Scene 4, Katherine and her gentlewoman Alice are practicing Katherine's English lessons, apparently in anticipation of further contact with the English:
KATHERINE Ainsi dis-je: d’ elbow, de nick, et de sin. Comment appelez-vous “le pied” et “la robe”?
ALICE “Le foot,” madame, et “le count.”
KATHERINE Le foot, et le count. Ô Seigneur Dieu! Ils sont les mots de son mauvais, corruptible, gros, et impudique, et non pour les dames d’honneur d’user. Je ne voudrais prononcer ces mots devant les seigneurs de France, pour tout le monde. Foh! Le foot et le count! Néanmoins, je réciterai une autre fois ma leçon ensemble: d’ hand, de fingre, de nailes, d’ arme, d’ elbow, de nick, de sin, de foot, le count.
The last lines, in my rotten translation:
"Thus say I: d'elbow, de nick, and de sin. How does one say "the foot" and "the gown"? [...] "The foot and the gown. Oh good Lord! These are words of a wicked, corrupt, coarse, and immodest sound, and not for honorable ladies' use. I would not pronounce these words before the lords of France for all the world. Fie! The foot and the gown! Nonetheless, I will recite my lessons one more time: d'hand, de fingre, de nailes, d'arme, d'elbow, de nick, de sin, de foot, le count."
The objectionable words, as Katherine helpfully highlights, are "foot" and "gown" -- at least pronounced to sound like "foutre" and "con", 17th century French for "fuck" and "cunt" respectively. But Katherine's not being depicted as an enthusiastic connoisseur of vulgarity dropping f-bombs and c-bombs all over the place -- her response is shocked and flustered upon finding herself as a gently-bred French princess using the language of the gutter, which in itself could be pretty funny. It's transgressive to hear her swearing, but not as transgressive as if she were swearing on purpose. French language is the butt of jokes elsewhere in Henry V, coming out of the mouths of both native English-speakers and ostensibly French characters like the leaders of the opposing French armies at Agincourt. The gag of heavily accented French characters mispronouncing English words so that they sound rude was enough of a crowd-pleaser that Shakespeare uses it in Merry Wives of Windsor with the comedy Frenchman Doctor Caius:
SIR HUGH If there is one, I shall make two in the company.
DOCTOR CAIUS If there be one or two, I shall make-a the turd.
So Katherine would not have been above some coarse humor because she was royalty, especially if the butt of the humor was Kate being French royalty. But the swears in her English lesson in Henry V are a little bit more multilayered -- she's mispronouncing English words so that they sound rude in French. Shakespeare's French may not be good French, but it's not "French-sounding gibberish" either. Why is there an entire scene in French in a play about Henry V, King of England, anyway?
Some of Katherine's mispronunciations are funny in their own right because they are homophones with English words, or simply because they are silly -- "neck" as "nick", "the elbow" as "d'elbow", chin as "sin". Hearing foreigners mangle the English language is a perennial feature of English comedy -- in the same play, Fluellen's Welsh-accented pronunciation of English words is mined for comedy, even though Fluellen himself is an ally of the English and a soldier in Henry V's army. Fluellen's presence and actions in the play always feels to me a little like the Elizabethan version of token multiculturalism -- hey look, there's a funny Irish guy and and a funny Scottish guy and a funny Welsh guy in Henry's army and they all love Henry too! It's a shoring-up of the discrete parts of Elizabethan British empire, as well as basic wordplay, and that may be the function of Katherine's French speech as well. Katherine is learning English presumably in (dramatic) anticipation of her marriage to Henry, but in the last act of the play Henry makes a big deal out of their language differences being no impediment in their union -- "O fair Katherine, if you will love me soundly with your French heart, I will be glad to hear you confess it brokenly with your English tongue" -- but also of his own lack of facility in either speaking French or wooing beautiful women. Katherine coming to learn English and to speak it, however brokenly, with her new English husband is in this way symbolic of France itself subdued under Henry's mastery. But all this comes after Harfleur and after abundant imagery comparing England's actions toward France to rape. This lends a slightly queasy undertone to Katherine and Henry's marriage, but also to the idea of cross-linguistic persuasion -- there's no reason for a rapist to bother speaking the language of his victim, or for a bloodstained conqueror to bother speaking the language of his unwilling conquest, but Henry's being nice, so Katherine coming around to speaking his language is a positive sign of her own acquiescence mitigating the necessity of future violence.
There's another union in the second tetralogy of history plays where the husband and wife don't speak the same language -- Mortimer and his wife in Henry IV Part One. Mortimer asks Glendower to translate between him and his tearful wife:
MORTIMER Good father, tell her that she and my aunt Percy
Shall follow in your conduct speedily.
[Glendower speaks to her in Welsh, and she answers him in the same.]
GLENDOWER She is desperate here, a peevish self-willed harlotry,
One that no persuasion can do good upon.
[The Lady speaks in Welsh.]
MORTIMER I understand thy looks. That pretty Welsh [the lady's tears]
Which thou pourest down from these swelling heavens
I am too perfect in, and but for shame
In such a parley should I answer thee.
[The Lady speaks again in Welsh.They kiss.]
I understand thy kisses, and thou mine,
And that’s a feeling disputation [...]
(I seriously doubt Glendower and the lady's spoken "Welsh" would be remotely as coherent to Welsh-speakers in the audience as Katherine, Alice, and Henry V's spoken French, given the Elizabethan penchant for foreign-sounding gibberish.) So this isn't quite the language of love overcoming national differences, but the idea of an English man with a foreign wife coming around to her foreign ways in the context of their union is a theme in these plays as they deal with both international conflicts and the development of English empire.
(Historically speaking, Henry V and his predecessors would have been fully capable of speaking French, even if it was semi-mutated French -- but Henry V promoted the use of English over French in courtly and governmental use, reaffirming a shift to Middle English vernacular that had begun earlier with Richard II.)
So the French dialogue, rude words and all, serves a rhetorical purpose in a play about the historical relationship between England and France that would not be served by French-sounding gibberish. Would the humor have carried over? Would the audience have understood the dialogue itself? The mispronunciations in Katherine and Alice's English lesson can be funny even if the rest of their dialogue is unintelligible, but the foot/gown bit relies on at least knowing enough French to recognize "fuck" and "cunt" and to grasp the gist of Katherine's long stream of flustered statements afterward. Of course, depending on how broadly these scenes are played, they might be funny even to audiences who speak neither French nor English, but would it have been common in Shakespeare's time for middle-class and above listeners to understand at least enough French to get the gags in Henry V? I don't know how common, and we don't have solid statistics for bilinguality in the general Elizabethan public, but these scenes and their content suggest at the very least that Shakespeare thought his audiences could and would get the joke -- enough of the audience to merit these scenes' inclusion. [1/2]