Eventually, such an Empire would have to shift towards France, or lose it. France was the more powerful and wealthy of the two, not to mention more populous. If they refused to eventually become more French centered, they would lose France, which was of more value at the time, then Britain.
I'm guessing that wouldn't be too hard. Some Plantagenet Kings of England were used to spending 2 thirds of their
regnum touring through the French duchies anyway. Either to ensure loyalty, suppress rebellion or fight off invasion.
A dual-capital system with the King holding court at both London and Paris could work,too. Even more flexibly the King's court could be held anywhere the King was at the time, though a city or large town obviously served better than a faraway village.
The first couple of generations of their rule, the politically savvy Plantagenets probably wouldn't even bother going through the maelstrom of uniting the kingdoms, I think. Why rush into things when you can take it slooooow...
As to the possibility of any of that happening, I have to admit, that reading through the history of the period, I always wonder why the Plantagenets didn't manage to capture the French throne!
Their power base wasn't essentially that much smaller than that of a French king. Additionally it wasn't as fragmented.
Even by that time when French royal demesne had expanded considerably, it was a small part of overall France and it was surrounded by a multitude of powerful duchies that had been known to be less than loyal, on occassion. The Plantagenets seem to have been experts at playing the ambitious dukes against their King, too.
So, it really is beyond me why they didn't manage to capitalise on their advantage over the French King during a centrury of fighting and plotting...