It needn't be political leadership to get the charge-up: being a religious leader or someone internationally famous/popular would also work, just so that lots of people are *focused* on you. The verb here is closer to "worship" than "focus". Princess Diana is probably the best example of a worshiped political figure in recent memory; JFK was another. Whereas Bill Gates is probably also exceedingly famous, but the focus he gets is likely not charging his Hegemonicity.
Possible charging methods: For the entire time, one bloodline is political ruler of the whole linked worlds, one is leader of a powerful religion, and one is an iconic entertainment figure. Or the third could be leader of a major institution (e.g. psi academy, galaxy's premiere research institute, interplanetary peace society). Either of these versions will get the job done, but in either case the third bloodline will have a hard time charging as much as the other two, because it's just hard to be as big a public figure as the President or the Pope. Also, it's harder to *control* becoming amazingly worshipped as an entertainment figure - there are an awful lot of entertainment figures who aren't particularly impressive at all. (It's also hard to create a powerful religion, but once you have it you can probably keep it in a way that entertainment-celebrity doesn't).
Potentially, the Superhero is the right kind of popular icon to draw worship-focus and successfully charge the bloodline. If you had a masked superhero that was as popular as Superman is in most comic book versions of Superman, then that would probably be sufficient to charge. If you had a masked superhero that was as popular as Spiderman is in most comic book versions of Spiderman, that would definitely not be. Not having any comic book masked heroes currently extant in the Linked Worlds, you don't know where a new one would fall.
(The Inopportune Moment crew are moderately popular with many people on Lendt, but probably not to the extent of charging Janina Megaera very much at all. Jim is probably a little closer to worship among the teeny-bopper crowd, but that's a narrow subset and also a little light.)
A potential plan: the Scouts!
If they existed, they'd be a) a public, well-known institution which could pretty easily win a place in the culture, b) an institution with memory and structures for enduring over time, c) an institution whose public image would (or could) be about serving the people, doing good, righting wrongs, helping those in need--just the sort of image that could inspire worship, for the same reason that a religious leader or masked superhero is the right kind of icon.
So, have the leader of the Scouts be a bloodline hegemon, and make it hereditary. (Maybe you'd want a structure where there was the Leader and the Chancellor, with the latter non-hereditary, but that's a question of principle rather than a question of whether it will work for charging the line. ) Part of the Scouts' charter, as an institution, Would be to ensure that the hegemon is worshipped (this is a PR task. Scouts seem to be good at PR. ). And to remember about the bloodlines plot.
Make the Scouts into a big public institution--almost like a philanthropic foundation, except that sounds like giving money to worthy causes, which isn't what I mean; but the idea would be that the Scouts would be this thing people would turn to in distress. The trick would be to make it so the hegemon personified the Scouts in the public imagination; rather in the way that the Elite were cast as extensions of the Hegemon's will, the Scouts would have to be seen as extensions of the great philanthropist.
This could work, but you might have to do something to visibly forge the bloodline hegemon and the Scouting organization together.