Not only a unified Frankish Empire (even if divided in quadrants headed by Elector types) but also a concerted power projection to take either Galicia without opposition from surrounding powers
Err... What surrounding powers?
A unified Carolingian Empire is going to totally DOMINATE any surviving neighbour, except for the Byzantines.
Just get the ERE to fall, or get reduced in power, as OTL, and this HRE would naturally expand, first into pagan lands, then quite possibly replacing the ERE as a supporting power against Islam.
For the Iberian Galicia, get one of the tiny remnant Christian kingdoms to swear fealty to the Emperor and join the HRE in exchange for helping the Reconquista, and whichever ruler that is will massively expand territory at the expense of the growth of the other small Iberian kingdoms. If this first kingdom isn't Leon (which, IIRC owned Galicia), then either Leon unites via marriage (as Castile did iOTL), or it joins the HRE to get more of the spoils.
For the Ukrainian Galicia, that might come as a result of a pagan kingdom (e.g. Poland) not only converting, but also joining the HRE. This kingdom then expands to take in the area of OTL's PLC, say - at least the southern portions, and we're done. Candidates other than Poland could include the Czechs or Hungarians. To KEEP this land, it would help to get an earlier version of the Unates (Byzantine Rite churches under the rule of Rome).