Why do you think the Pi is powered via the inverter? I can't see mention of this, and the picture of their setup shows the Pi being powered by a step-down from 12 V.
edit: His desktop computer is powered by this at times. Not pictured and not labelled on the photo. Makes sense though.
Maybe you tripped over the text "So I decided to build a new setup inspired by Will Prowse solar demo setups, which is pictured below:", which I understand as "So I decided to build a new setup, pictured below, which was inspired by Will Prowse solar demo setups:".