A couple of books for you perusal. available on Amazon
(other booksellers are available)
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Not the origin of the ‘bale out’ term of course, which likely predates that.
Last night I looked here and there (including the Internet Archive, which has OED, but I think it’s the 1913 edition, which might not say much about aircraft stuff), and concluded the following (even though I cannot boast of Acadian heritage): me, I don’t know.
But Joseph Heller,* a North American 
, seemed to prefer the ai spelling.
I can’t help but wonder–just wonder, not yet believe–whether the sense in question involves escaping the confinement (i. e., the figurative jail) of the aircraft. Again, just wondering.
Anyway, now I can’t get this song out of my head:
*or his editor(s)/proofreader(s)
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Oh. Well then we’re back to it making no sense to spell it like a hay bale.
But not just hay comes in bales, you can have a bale of wood, bale of paper, bale of clothes, bale of cotton, bale of cloth etc. all of can be tossed out of an airplane. Cargo was often baled up for transportation.
There is also the archaic meaning of bale,
noun Archaic. evil; harm; misfortune. woe; misery; sorrow.
If you do have the misfortune of having to bale out of an aircraft, you may be full of sorrow.
Just another one to add to the mix, if a business was going bad, would you.
“bail out” or “bale out”? would you stay or would you go?
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Only if “we” are American, and, since we are not, we can’t go back, since we were never there.
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Gareth and Christian
yes you can throw them out an aeroplane too
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Jump down, turn around, pick a bale o’çotton
I was assuming “we” to be rational thinkers and not nationalist nitwits. Apologies if that was too much of a reach.
Of course. I understood the previous poster to be saying the use of the term “bail/bale out” predates the use of airplanes for such things, though. Then again, I suppose bales were also tossed out of wagons.
F@ck that little mouse,
Cos I’m an Albatraoz.
Back to you. 
See rationale above. Havn’t seen any equivalent rational defence for the water imagery, other than alleged historical precedence.
In British English there doesn’t seem be a need to decide, since both seem to be in use.
If Americans have opted for one over the other, its not very surprising that they opted for the wrong one, especially if they thought of it first.
Uh?
What if a sheep behaved baa baa-dly and was arrested and the British bulldog magistrate required said sheep to relinquish a quantity of woof woof wool? Released on bail or on bale? The sheep’s in the dock, so probably the former? 
Now you’re just talking bollocks.
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My bollocks are no match for yours 
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Just trying to get a bit of free form poetry going. Not to worry. 
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It’s the movie tag line we’ve all been waiting for. But what’s the film?
I know what Ewe did last summer 5
Other possible titles include:
Ewe only live twice.
Catch me if Ewe can.
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I like @tempogain’s link above because it’s the only argument that examines all the uses of the different terms/meanings.
(emphasis mine)
The American usage of just getting out of an undesirable situation, not specifically a plane, was the one the OP used which caused all this hullabaloo, so bail is probably correct.