A "battle of the sexes" implies a conflict between a man and a women, or men and women, regarding gender roles in a given environment or circumstance, or a more generalized battle for supremacy between men and women.
There are no general battles for supremacy in any of Shakespeare's plays. There are no all-the-women-in-a-play versus all-the-men-in-a-play battles, as there is in Lysistrata, an ancient Greek comedy written by Aristophanes, in which the women of Greece agree to withhold sex from the men until the men end the Peloponnesian War between Athens and Sparta. The men agree to end the war, but once this issue is resolved, the men and women return to their "assigned" and accepted roles in Greek society. Who won that "battle"?
In Shakespeare's plays, the "battles of the sexes" are highly localized, which is to say that they occur only between individuals, in a clearly defined environment, and on a very limited scale. These "battles" also occur with the environment of Elizabethan and early Jacobean societies and are therefore subject to the societal norms in effect at the time the plays were written.
In A Midsummer Night's Dream, the only man-woman conflict is between Oberon, King of the Fairies, and Titania, Queen of the Fairies. They're squabbling over who will have the companionship of a changeling boy, an Indian prince. That's the extent of the "battle of the sexes" in A Midsummer Night's Dream. There's no gender-role conflict, or any supremacy issues, other than who gets to keep the Indian boy.
While it's true that there appear to be issues of a male-dominated society between Hermia and her father, Egeus (backed up by Theseus, Duke of Athens), regarding Hermia's desire to marry Lysander, this isn't a gender-based or generalized men-versus-women conflict. A young woman wants to marry a young man of whom her father disapproves, in a society in which a father's approval was necessary for the marriage to take place. Hermia and Lysander find a work-around solution to the problem, and they all live happily ever after.
The Taming of the Shrew is often cited as the ultimate Shakespearean "battle of the sexes." It's not, in my view. The battle between Petruchio and Katherina is one of personalities, not gender or societal norms. Both Petruchio and Katherina reject societal norms, so that's not an issue. By the end of the play, it appears that Petruchio has "tamed" Katherina, but only because Katherina let herself be tamed, and Petruchio was no less tamed by Katherina than she was tamed by him.
There's no "battle of the sexes" in Macbeth. It appears for a time that Lady Macbeth dominates Macbeth emotionally and browbeats him into killing King Duncan, but they both want the same thing: for Macbeth to be king. No other man-versus-woman or men-versus-women issues arise in the play.
This is likewise true in As You Like It, Romeo and Juliet, and King Lear.
Gender-wise, As You Like It is wholly conventional. Rosalind and Celia work within the prevailing societal norms to get what they want.
Even so, Rosalind dressing up like a man is a safety issue, not a gender issue.
ROSALIND. Why, whither shall we go?
CELIA. Alas! what danger will it be to us,
Maids as we are, to travel forth so far?
Beauty provoketh thieves sooner than gold.
CELIA. I'll put myself in poor and mean attire,
And with a kind of umber smirch my face;
The like do you; so shall we pass along,
And never stir assailants.
ROSALIND. Were it not better,
Because that I am more than common tall,
That I did suit me all points like a man? ...(As You Like It, act 1, scene 3)
More to the point, Rosalind (a woman) uses her disguise as Ganymede (a man) to manipulate Orlando (a man) into learning how better to love her (Rosalind), and into become a man more to her own (Rosalind's) liking.
Is there a men-versus-women conflict in Romeo and Juliet? Is Juliet at odds with Romeo because of their gender roles? The societal conflict in Romeo and Juliet is the "ancient grudge," the unending feud, between Romeo and Juliet's families. If the Capulets were all female and the Montagues were all male, then there might be some basis for suggesting that there's a "battle of the sexes" in the play. but even then, the conflict as it stands in the play would still have nothing to do with gender roles.
A test of whether a "battle of the sexes" exists in King Lear is to change Lear's daughters into Lear's sons. Clearly, it makes no significant difference to any issue in King Lear if Lear's three children are men or women, and there are no other gender-related issues in the play.
Also remember that during Shakespeare's time, and until the Restoration in the early 1660s, all of the women's roles in Shakespeare's plays were acted by men, so there's that issue to consider as well.