Feudal centralisation is a deep contradiction.
It just can't happen : feudality is based on wide distribution of power and while it can be unified (bureaucratically or dynastically), the moment the system gets centralized it simply stops being feudality.
Even late "feudalism" (to be understood as a rather diverse ensemble based on similar premises) was still based on personal links and exchange of services, where state is made from the different directly tied to an important noble or communautary structure (as in medieval city-states) and while organized, still defeated imposing its full bureaucratic authority even on directly relevant entities.
What you'd need would be a bureaucratic state, with more or less strong legacy from classical feudality, not unlike late Valois or early Bourbon France.
That said, I think you're mislead : on several things, which may need to be more well understood in order to fill your OP.
Polish nobility : generally, medieval nobility tended to be more important than generally tought (while we're talking there of all nobility : palatial nobility, high nobility, petty nobility). At the notable exception of England, it could range from 4 to 10% of the population, and while rare, the latter proportion was the polish one, but not only.
Calling nobility a negligble class is simply blatantly wrong for most of the cases.
Of course, we're including szlachta there as well.
A second point is that centralisation leads almost always to absolutism and remove popular "participation". But feudal settings didn't have much popular participation, save locally and these almost never disappeared, but were swalloed up.
A strong polish kingdom isn't inherently opposable to local autonomies : the point is that polish kingship should be the arbitle and then the autority over these autonomies.
A quicker Piast political renaissance after the XIIth crisis, critically in face of German dynamism, is definitely needed too, while I'm not too sure at which point it could be made; and that will pass by butterflying away Mongol Invasion of Europe.
Such political renaissance wouldn't be a return to "good old days", but would be at least an aggiornamento regarding Polish structures.
It would be wise to prevent Poland to expand too importantly in the East (too much concessions were made to szlachta for the sake of it); while such expansion is still definitively needed in order to strengthen alliance between the king and petty nobility over high nobility, as these lands should be trusted to the formers.
A definitive integration of Pomerania is such most needed, and maybe a more confrontational relationship with Lithuania.
Basically : don't try to copy/paste what happened elsewhere in Europe, and don't think that only what managed there can be applied, but try to find something that would answer the particular "needs" of Late Medieval Poland. The first step is to allow Poland to not being in continual crisis in the XIIIth century.
