portable washing machine and the leisure battery (https://i.stack.imgur.com/MeQi6.png)
12 V times 120 Ah gives 1440 Wh. 1440 Wh/300 W gives 4.8 hours. Figure about 4 hours since your inverter will not be 100% efficient. Reduce that to 2 hours, since you don't want to discharge your batteries too deeply.
Assuming your 300 W represents the power needed by the agitation motor, you need to be aware that motors have a big turn-on current surge, typically about 3 times running current. So peak load is about 900 watts. If the inverter can handle this (and you need to verify this) then at 220 volts this is 900/220, or about 4 amps. Multiply by 220/12, and you get a starting surge at the battery of 73 amps, which is well below your battery rating.