So the smaller header tanks that feed the engines during the landing burn need a better pumping system to ensure steady flow of fuel during those last few seconds, even when the rocket is re-orienting from horizontal to vertical - a move that would cause fuel in ordinary fuel tanks to slosh around.

That is a tough one. Aerobatic aircraft have a similar problem:

"To ensure the flow from fuel tank to fuel injector, aerobatic aircraft with the fuel tank in the fuselage have a “flop tube,” a flexible hose with a weight in the free end, plugged into the fuel tank. In normal flight, the weighted end of the hose flops to the bottom of the tank and draws fuel from there. When the airplane rolls inverted, the weighted end flops to the top of the tank, with the fuel. Regardless of the aircraft’s attitude, fuel and flop tube end up in the same spot."

https://www.airspacemag.com/flight-today/how-things-work-flying-upside-down-27746739/

When using cryogenic liquid oxygen and methane as fuel, this solution might not work so well, since rubber spontaneously ignites in the presence of pure oxygen. Maybe a steel braided hose, like for a washing machine. But hard to say. SpaceX has probably worked through a lot of designs already.