anglefire That graph looks pretty familiar, although I don't get as much of a clear spike as the three smaller ones you do (probably the sample time missing the motors starting up). The load looks similar though except we run our dishwasher earlier in the evening (when we're still generating enough to run most of the cycle).
2.4kw definitely seems too much still - my guess would be an immersion heater or an aircon (I'd imagine a really chunky heat pump might spike that high but the time of year makes that less likely anyway). Given the confusion from the OP I'm guessing though that anything obvious like that would have already been noticed.
If anyone has more experience with appliances, would an older fridge/freezer (one which might be in need of replacement) use more power trying to cool? I'm wondering if wear and tear might account for a higher spike load. I'm thinking a problem analogous to needing to re-gas the aircon in a car due to failing performance.