Honestly, @WelshCamper's spot on about smart load management being the key here. I've got a 5kW system feeding my garden office and ran a basic washing machine for a while—worked fine if I timed it right.
Your real constraint isn't battery size, it's inverter rating. Most budget inverters panic with the inrush current when a motor kicks in. You'll need at least 3-4kW continuous rating, probably 6kW peak capacity to handle the startup spike safely.
What actually worked for me: run the machine mid-morning when solar's generating properly, not drawing from battery. You'd be surprised how often that's viable in UK summers. Winter? Different story entirely.
With 6kW nominal you're in decent territory, but check what your actual usable daily output is (accounting for angle, season, your location). A decent MPPT controller makes a difference too—Victron stuff is pricey but bulletproof.
The inductive load point @AngliaOffGrid mentioned is real, but mainly just means don't cheap out on the inverter. What's your current inverter spec?