To run a motor from a sine wave inverter is problematic.
The inverter is making a sine wave at about 8 Volts using transistors switching on and off to create a sine wave shape to mimic the mains supply.
The 230 Volt output is derived from this 8 Volt sine wave by a ‘step up’ transformer. To make an inverter with a 2000 Watt capacity, (2000W/230V = 8.6 Amperes) and to make this current capacity from 8 Volts (230/8 = 29) that is 29 x 8.6 = 249 Amperes.
The inverter then is drawing 249 Amperes from the batteries, via the inverter to support a 2000 Watt load.
The washing motor is a commutating motor, thyristors switch on & off to regulate the speed of the motor at 100 times / second, each switching event turns the supply on creating both a current surge and Voltage spike in excess of the 230 Volt supply from the inverter.
In normal domestic installations the supply is softened by the transmission line which can run for many miles, spikes and surges are absorbed.
In a boat or RV, the transmission line may be just a metre, with no room to absorb the interference created by the motor, supply impedance is very high, the interference is like hitting a brick wall.
The inverter is being ‘hit’ by these switching events at the output transformer @ 249 Amperes & some 300 Volts peak.
Whilst the light bulb may be absorbing some of these switching events, a solution may be found in decreasing the circuit impedance, a trial fix may be found by hiring a ‘site’ 230V / 230V isolation transformer form the hire shop. The transformer comes with a fixed 13 A socket and cable with 13A plug, very heavy.
Here I have to come ‘clean’, I do not have a washing machine on board and cannot test this, I am hoping this will help to get things working. A day’s hire may lead to a permanent solution, and possibly an answer?