I think you basically answered the question yourself near the end - you give the Church the role you need to balance the inherent inequality of the feudal system, just don't have the King as the Head of the Church - you'll probably need a constitutional block or something to keep him in his place.
To the average commoner in that type of society, King and God were both pretty much on the same level anyway - all-powerful figures you disobeyed at your peril, lest their more earthly agents come for you.
Imagine if the Catholic Church in medieval Europe was a bit more pious and benevolent than it was in reality - with that much power and influence, how much do you think they could have fulfilled the role you describe? You could look at King John and the Magna Carta, and imagine if that was driven by this kindly Church instead of rebel barons?
Keep your feudal system, keep your feudal professions, keep your regional lords and barons, keep your rustic rabble, just tweak your religion.
