If it is already struggling at 50%, which would be 1200w, then doing the on-off trick to pre warm the elements is very unlikely to work for the 100%.
The hunting and surging at 50% informs what the expected capacity could be as things are right now and will give a more honest indication than the rating sticker. Did you have to do some tweaking of the throttle/carb/choke etc to get it running?
It's rpm may be too low.
The situation with the washing machine is consistent with a low rpm from the generator.
Sorry I can't give you a definitive answer to all this and I know it's frustrating when gear doesn't work as per how it should.
Without being there in person with Multimeter in hand, fault diagnosis becomes mostly informed guess work.
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