The only viable option I have ever seen is, in fact, Charlemagne marrying Irene and producing an heir
The whole proposal doesn't looks entierely credible : it's mentioned by Theophanes the Confessors IIRC, in an hostile outlook against Irene. It simply doesn't appears in Carolingian sources.
Assuming this was genuine (and I don't think it was), there's no way Byzantine court would let her do this : if we consider Theophanes' account, she was chased of power as soon as it was out of the bag.
There's a possibility that it was mixed (willingly or not? It might be a case of rumor getting written down too) with the proposed union of Rotrude and Constantine VI which seemed to be workable enough that Rotrude was sent to Constantinople she was taught greek by a Byzantine monk, until the whole thing broke off in 787.
Peventing this matrimonial project's abandon is doable, not especially obvious, but doable : it wouldn't give much in the way of an imperial unification, this being said.
(and agreeing to drop Frankish inheritance law and proclaiming that heir the undisputed inheritor of the re-united Empire)
Until the death of Pepin of Italy in 810, the imperial title of Carolingians wasn't really supposed to outlive Charlemagne. Louis being sole survivor and steeped in a clerical culture putting the stress on a Christian imperium did changed things on this regard.
Note that "Frankish inheritence laws" (which we probably agree shouldn't be confused with actual inheritence customs) were an evolution from late Roman practices of shared/split rulership (as co-imperialship in Constantinople evolved from, even if without the territorial part) that was expected by Latin societies (not just Frankish, but Italian or Aquitain as well, for instance).
It would be hard to get rid of something ingrained in institutional conceptions since centuries in most of Latin Europe : historically it asked for the collapse of Carolingia, and it did survived partly into the custom of giving the leading title to the heir before the death of the king (as in France before Philipp II) or duke/count/etc. (such as in Toulouse until the end).
That Carolingia was, eventually, a relatively weak state overall (even compared to Merovingian Francia in some regards) that was built right from the beggining trough aristocratic takeover and support (Carolingian having put great effort building up first forms of feudalities) doesn't make me things that "dropping" strong institutional customs and pulling a centralized empire would be doable.