APU Fuel Path
The APU is fed from the left engine fuel manifold, which is supplied by either left boost pump (main tank aft, main tank forward, center tank left), or by an additional DC pump in the left main tank. When, as the AAIB said they did, the boost pumps are working, there is positive pressure in the fuel manifolds.
Starting up the APU (for which there is no evidence) would not lead to a sufficient pressure drop to starve the main engines. The APU fuel demand is tiny compared to the supply capacity of the boost pumps.
Bernd