Searching for a literary term for “if this, then I’m a this” statements

I’m reading Shakespeare’s Henry IV (Part 1) right now and I’m noticing that Sir John Falstaff has a propensity of saying “If this, then I’m a this” sort of statements. A few examples to clarify:

… An I
have not ballads made on you all and sung to filthy
tunes, let a cup of sack be my poison

An I have not
forgotten what the inside of a church is made of, I
am a peppercorn, a brewer’s horse

if manhood, good manhood, be
not forgot upon the face of the earth, then am I a shotten herring

If I do not beat thee out of thy
kingdom with a dagger of lath, and drive all thy
subjects afore thee like a flock of wild-geese,
I’ll never wear hair on my face more

Is there a particular literary term for these sort of sentences?

I would call it a "rhetorical conditional" which is amplified (or exploded) by use of hyperbole and reductio ad absurdum.

A standard conditional:

If it doesn't rain tomorrow, then we can have a picnic.

The previous sentence simply states a condition and provides an expectation.

A rhetorical conditional:

If you be men, then you will take up arms and fight like men.

This sentence uses the simple conditional as an exhortation, something that may inspire or shame or otherwise move the listener. It is also an example of understatement.

Now, as Shakespeare has Falstaff using it, it becomes more complicated. Falstaff is a grand comedic character, fully flown with alcohol and oratory, and he seems never to speak except in pursuit of a kind of amusing rhetorical excess. His discourse is marked with a habit of "reasoning things out" while coming to exactly the conclusion that feeds his fancy. Listen to him here in Henry IV, Part One as he works out for himself how to take credit for defeating Percy, even though he was hors de combat, feigning death, while the deed was in fact done by Prince Henry:

FALSTAFF
[Rising up] Embowelled! if thou embowel me to-day,
I'll give you leave to powder me and eat me too
to-morrow. 'Sblood,'twas time to counterfeit, or
that hot termagant Scot had paid me scot and lot too.
Counterfeit? I lie, I am no counterfeit: to die,
is to be a counterfeit; for he is but the
counterfeit of a man who hath not the life of a man:
but to counterfeit dying, when a man thereby
liveth, is to be no counterfeit, but the true and
perfect image of life indeed. The better part of
valour is discretion; in the which better part I
have saved my life.'Zounds, I am afraid of this
gunpowder Percy, though he be dead: how, if he
should counterfeit too and rise? by my faith, I am
afraid he would prove the better counterfeit.
Therefore I'll make him sure; yea, and I'll swear I
killed him. Why may not he rise as well as I?
Nothing confutes me but eyes, and nobody sees me.
Therefore, sirrah,

[Stabbing him]

with a new wound in your thigh, come you along with me.

[Takes up HOTSPUR on his back]

Notice the conditional in the first part: "if thou embowel me to-day, I'll give you leave to powder me and eat me too tomorrow."

It is a rhetorical conditional, but it benefits (as do your examples) from pairing a ridiculous, hyperbolic consequence with the condition. There is no way in hell Falstaff would consent to be powdered and eaten, nor is it likely that anyone (much less Prince Henry) would be likely to take him up on it. In the example you cite, Falstaff and his audience would agree he could never be a "shotten herring" or a peppercorn, so they (and we) would expect the conditionals to be set up only to be shot down by his reductio ad absurdum.

Addendum

There is actually a rhetorical term for "rejection of an argument by means of a ridiculous comparison" — diasyrmus — which could probably fit here as well if one considers a conditional to be a type of comparison.