Where does the phrase "get a bye" come from?

Solution 1:

In live hare coursing, a bye means to pass to the next round due to a missing opponent, but not before the dog has made a lone run so it doesn't have a rest and an unfair advantage over the others.

The National Coursing Club's glossary says:

Sometimes runners are withdrawn from their courses, either because of absence, injury or weariness. Their opponent still has to run a course – a “bye” – so that it will have run the same number of courses as the next opponent. The dog may run alone or accompanied.

So in the first example above, greyhound Bismarck's opponent had pulled out, meaning he could progress to the next race, but not before having to run a punishing run to "even the playing field".

And in the second example, the day ends with greyhound Duffer running a bye on its own.

And again in the third example, where it's clear the opponent was injured the day before.

We can see bye used a lot in any of these books, and "run [...] bye" in many other coursing books. As can be seen, "get a bye" isn't in itself any kind of special set phrase (at least in these examples), but just some words used together.

Solution 2:

The usage of bye illustrated in your examples is still current in dog events. The meaning is made clear in rules #14 and #19 of the Tulsa, OK bird dog club:

14. A handler must attempt to hunt this dog competitively for the full 30 minutes or if a dog is off the course for more than 10 minutes then the judge will stop the brace. A bye dog will be selected by the field marshal and will run the remaining time left of the brace.

19. Any participant which is absence at start of his brace will be allowed ten minutes to show or will be automatically disqualified for that brace and brace mate will run with a bye dog. A bye dog must be selected by the field marshal and will receive no score.

Thus, "Bismarck ... had the misfortune to get a bye of terrific length" means that Bismarck had to fill in for an errant or missing dog, which wore out Bismarck before his or her own race.

Although this makes the meaning clear (and that that meaning is not entirely at odds with the conventional meaning, a pass), it does not resolve the question of why the word is being used in two different, confusable ways.