I'm afraid this post will be somewhat inconclusive. If you want something definitive then follow this link for yet another review of The Horologicon.
Once upon a time there was the word bit. But not in the sense of a small piece of something. This bit meant bucket, and was cognate with bucket and with a water butt. Somebody who was in charge of a bucket of water was therefore a bitter. So this became a term for a fireman. As in this line from 1467.
That the bitters be ready with hur horses and bittes to bring water.
But then a bit also came to mean a sturdy thing on a dockside that you could tie a rope to. And the end of the rope that was around the bit? Here's a quote from Captain Smith's Seaman's Grammar of 1627 (that's the same Captain Smith that had a thingy with Pocahontas).
A Bitter is but the turne of a Cable about the Bits, and veare it out by little and little. And the Bitters end is that part of the Cable doth stay within boord.
This remains in every dictionary to this day. Here's one from 1725:
BITTER, any Turn of a Cable about the Bitts is called a Bitter; and 'tis used that the Cable' may be let out by little and little. And when a Ship is so stopp'd by a Cable, they say she is brought up to a Bitter. Also that End of the Cable which is used to be wound or belayed about the Bitts,' they call the Bitter End of the Cable.
And that's what's in the OED. So the bitter end is simply the end of the rope. Right? Nothing whatever to do with bitter as in acrid and sour-tasting.
Well, that's what I wanted to say, but then I found this from a poem by George Wither published in 1622:
With hunger parched, and consum'd with heat,
I will enforce them to a bitter end;
The teeth of beasts I will upon them set,
And will the poisonous dust-fed serpent send.
Now it's possible that George Wither was just being nautical. But in the poem that's God speaking and so it's much more likely that he was remembering the fifth chapter of the Book of Proverbs:
For the lips of a strange woman drop as an honeycomb, and her mouth is smoother than oil: But her end is bitter as wormwood, sharp as a two-edged sword.
So now, well, I just don't know any more. All the authorities say that working to the bitter end is just getting to the very end of the rope. But it also seems to have combined with the more usual sense of bitter to make poor bloggers like me not have a proper conclusion.
The reason that I was researching this all in the first place, is that I've been reading Bum Fodder, an Absorbing History of Toilet Paper by Richard Smyth. It's a great little book and ideal for reading on the... yep. Anyway, he says that the bitter end was placed in a bit (or bucket) and was used by sailors for wiping. The OED says that the bitter end was sometimes used as a whip, which seems downright unhygienic. Frustratingly, neither provides a quotation.
Anyway, back to that review.
Everyday life with a dictionary writer.