Calling the Wasserfall engine the P IX is an error, possibly introduced by Roland Pocock in his 1960s book on German missiles. According the German wartime documents, P IX is the static test stand for both the Wasserfall engine prior to installation and the all up missile, with facilities to measure and adjust thrust, fuel flows etc. None of the german documents that I have translated ever refer to the Wasserfall engine as P IX, only the test stand.
The Brugge 2010 diagram above is misleading in that it shows a single internal layout for both Wasserfall and Hermes A1. It wrong for Wasserfall as, for instance, the big spherical pressurised nitrogen tank is missing. All Wasserfall variants build or conceived had this. I cant really say if its a good fit for Hermes A1 as the Americans did a much better job of archiving the German design drawings than they did for their own. Calling the Wasserfall the "German W5" is also an error. Wasserfall was usually called Wasserfall, occassionally (usually by outside agencies) its older Roth design office C2 designation was used. The W3, W4, W5 etc. were batch built variant used (or intended for) the uncompleted flight test programme. W5 was just a glimmer in the meeting minutes when Peenemunde was evacuated. The error of calling the Wasserfall W5 (so beloved of model kit manufacturers) and confusing it with the never built operational variant was introduced in the post war allied intelligence documents.
I think its also a gross oversimplication to consider the Hermes A1 as a "US Wasserfall". It has virtually the same (but not identical) aerodynamic form and control fin/jet vane layout. But different engine, different fuel, different oxidiser, different fuel regulator, different materials, different internal layout, different gyros, different flight control computer, different servo actuators, different guidance link transmitter & receiver and different guidance system. I don't know if changed to use Imperial threads and fixings but would love to know. So while the Hermes team struggled to get Hermes A1 off the ground, its not as simple as they were struggling to "make Wasserfall work". They were struggling to make a pretty much wholly new but Wasserfall shaped missile work, shortcutting the amount of wind tunnel time needed by re-using the aerodynamic form.