I found a thread on CPF
here states that the tail cap current of the Fusion Showerhead is 1.42 A. While I don't have a discharge curve for the Energizer Ultimate Lithium AA at 1.42 A, the
Energizer spec sheet shows a 1 amp constant current discharge curve, and it is clear that the voltage drops quickly and plateaus at about 1.4 V. At 1.42 amps, I would expect the voltage to be closer to 1.3 V under load. So I would expect that the voltage on the battery pack under load would be between 7.8 to 8.4 V.
Based on that analysis, my guess is that Energizer Ultimate Lithium batteries would be perfectly suited for this application.