Yes, it handles both the motor and the inverter: take a look at the description under the picture on their website:
Motor
IMO one of the problems of cooling the battery is that one should know where it heats up the most. I assume that that sort of polystirene of the skidplate hinders heat dissipation and that the cells close to the inverter are warmer, too. It would be nice to use an IR laser pointed thermometer like TCMB371 did in some of his videos to find out where the battery case is the warmest and target those areas more than others.
I also guess that the electronics of the bike knows the temperature of the cells and that in theory it would be possible to show the data in the app, like a thermal map of the battery case.