I have recently purchased a second hand A80 and Alesis NanoPiano module. As I cannot tune the module, is it possible to alter the master tune of the keyboard? So far I have only managed to transpose the keyboard. Am I the only one who finds the manual hilariously unclear?

Alesis manuals are way behind Roland concerning advanced use.
And I suspect that tuning the Alesis calls for the use of SysEx
I will recomend that you also (you might allready have done it) post a question in the Alesis thread concerning which SysEx that change the tuning.
At the same time you should also investigate wheter the A80 is capable of sending costomized SysExCommands. Good luck