In Chapter 5, the Bygone fight is easily the nastiest part of the game. The whole thing is RNG dependent, and if you fail (very likely), you'll be stuck in a very nasty puzzle that doesn't even tell you its a puzzle in order to progress. Here's how to get through it.

1) Getting beaten once by the Bygone is actually good. You'll unlock an event which conserves Ollie's energy which you'll need for unlocking the 5th ending.
2) During the first round with the Bygone, once you've hit any of its parts for the first time, take a look at the left part of the screen. A shadow will appear which you can interact with. Who knows what it's for but its very likely a flag for future routes. The Bygone will get a free hit. Don't worry. Like I said, its best you lose the first round.
3) Click on "Olfactory" once you lose.
4) On the second round with the Bygone, you have better odds of beating it since you go first. Aim for the wings, then gun, then sensor, then head.
5) Try not to lose to the Bygone three times in a row. Ollie will lose energy if you do and you'll be locked in the worst ending.

From a gamer who had to ride down that stupid elevator a million times and discovered the smelly answer by accident. There were no guides available on the net about this part...until now.