Assuming we're looking for in-universe answers... There have been times with the bloodline has been broken before. The reign of the Tsaesci Potentates is one example. There was another time before that, though, which hints at how things may have played out...
The Alessian order was destroyed during the War of Righteousness in 1E 2321. Presumably, this broke the bloodline from the original convenant between Akatosh and Alessia. What happened to the Amulet of Kings? Who knows. Why didn't Dagon invade back then? Who knows. It is worth mentioning, though, that these were not good times. The Empire had dissolved and the people were suffering from a plague and a nasty civil war. Perhaps, instead of invading with a corporeal army, he used his influence to sew destruction through other means? (Source: http://www.imperial-library.info/history/1.shtml)
Four hundred years after the War of Righteouness, the creation myth of the Reman dynasty asserts that the spirit of "El-Estia, the queen of ancienttimes"--whom I take to be Alessia--gave the "Chim-el Adabal" (the Amulet of Kings) to an infant Reman I to announce the coming of the second empire of men. My assumption is that this "resets" the bloodline, so the bloodline of interest is now the Reman bloodline, rather than the Alessian. Of course, it's unclear how much of this story is true, and how much of it was made up by the Reman dynasty to legitimize itself through the legend of Alessia. (Source: http://www.imperial-library.info/obbooks/remanada.shtml)
Of course, the Reman dynasty didn't last long. The events of 1E 2920 ended the Reman dynasty and started the Interregnum, where the Tsaesci Potentates from Akavir ruled the Empire. Once again, it is unclear what happened to the Amulet of Kings, but there is no report of Mehrunes Dagon invading during this period either. However, in the middle of the Interregnum (2E 283) there is another long and deadly civil war. Perhaps this war is due to the destructive influence of Mehrunes Dagon, or perhaps it is due to the presence of an alien administrator.
The http://www.imperial-library.info/obbooks/remanada.shtml mentions that the Amulet of Kings was lost again during the Interregnum. It begins to talk about the rise of Cuhlecain and the third empire of men, but it is not finished. Presumably, this chapter would have described how the Amulet of Kings was returned to Empire for a second time.
Regardless, it provides a possible answer for what happens to the Amulet of Kings when the bloodline is broken. It stays gone for a while (about 400-800 years, it seems) before being put in the hands of the next Emperor through some supernatural means. So there you go, a semi-plausible explanation for historical inconsistencies in regards to the "bloodline" of the Amulet of Kings.
Very nice, I like that a lot.
Let me just add a few thoughts. I don't think bloodline is the perfectly correct word for the whole issue, because bloodline creates the image of a descendence from a line of kings which is based on physical heritage. That is, imho, just one part of the whole deal. There's the mystical/divine aspect, too. But that's semantics, so I'll leave it at that.
You've got the fixation points of the whole story right on spot: namely, there are three:
1.The creation of the original pact (First Empire) between Alessia and Akatosh (or Lorkhan, depends on who you ask or which source you read (Trials of St. Alessia or the Pelinal-ada, last chapter)).
(Interregnum) - which is described as a not so nice time
2. Recreation of the First Empire: initiated by King Hrol (Bonus question: What did he do and which connection does he have with what Alessia did/represented?)
(Interregnum by the Potentates) - also, from stories, not the best times to live in
3. Another recreation followed: the Septim Empire. There are sources about the relation between some participants (which may or may not be the same person): Hjalti, Talos, Zurin etc. Now the real thing imho to think about comes up: the reestablishment of the divine right to rule ("mandate of heaven"), which means the Dragon Pact (or/therefore: fitting oneself into an existing framework of events/rules).
I postulate (been done before, by others, too): All three events share basic things that have to be done to reestablish a line of kings, including the mythical connections.
In simpler words: If you want to try for a new line of kings, it's best to start making love to a hill (or what or who it represents). If you have what it takes, perhaps it works.
(Comment: anology to real-life mythology: If you wanna be king, try and pull the sword out of that rock. In TES, it's just the other way round. You put a sword into a rock. If someone detects Freud he or she may keep him.)