Along these lines, the only way by clan custom to decide which Jade Falcons are the 'legitimate' is basically a trial over it.
Regarding if Alaric and Stephani Chistu will 'deal' with Jiyi's... I feel like Alaric will just declare they answer to Stephanie Chistu because she answers to Alaric. If Jiyi says nope, then it'll arcs towards violence, because Alaric is a guy who in June 3152 declared he owns the universe. Not really any room for compromise with the guy as he exists now.
But... unless Wolves power up like saiyans again, or Stephanie's Falcons do so, the Falcon factions aren't likely to meet each other directly any time soon.
Alaric's message is timewise quite hilarious considering it came right after the Empire's capital had been sacked and a sibko of his children was abducted. I doubt the Dragoons will send a few of those to the Hinterland Falcons but it would be funny nontheless.
Also, he can delcare the Terran Falcons the real Jade Falcons all he wants but the hard relaity is that there are at least two factions that claim to be Jade Falcons that don't answer (for now) to the IlClan. The main question will be if the "IlKhan's bodyguard" will even be allowed to travel back to their old territory to get access to their own genetic legacies. I reckon that will take some time unless we get a Deus ex Machima that says "The two Terran Clans were so smart they brought a full set of their genetic samples to Terra and kept them out of arms reach". After all the Wolves have the same issue but on the flip side have a vastly bigger gene pool left (surviving warriors) to start again