The pen enabled intuos range of tablets use electromagnetic radio waves in order to power the pen and work out position/tilt/pressure etc. Unfortunately an inherant property of an inverter is to pulse current through coils very fast in order to boost a voltage. It is this pulsing of current that causes quite strong radiowaves to be transmitted from the coils.
The trouble is that because the radio waves created by the inverter tend to be sharp abrupt and spikey, they can easily mix with the waves present between the tablet and pen and in the case above cause random double clicks. There's not much you can do accept maybe put the inverter inside a Faraday cage of steel mesh or move the inverter at least 2 metres from the tablet.
More info on this can be found in the forums at www.bongofish.co.uk where enthusiasts such as myself hack these tablets to be similar to the Cintiq range.