I worked with generators a good deal in the Army Signal Corp. A generator will be most frequency and voltage stable and provide the best regulation if you have a modest "base load". This would be especially true of a station operating CW. I would suggest a small base load of 100 to 200 watts in a purely resistive load like a lightbulb. Your 2KVA genset should still be quite happy as the max load will still be well under ratings and CW is about a 40% duty cycle.
For whatever it is worth: I have a 6KW portable genset and I have found that petrol quality is a factor in frequency/voltage regulation. I can't explain why, but I do know from some tests that frequency/voltage stability is best when I use premium NO Ethanol fuel. The difference between 87 octane with Ethanol and pure petrol 91 octane was not huge, but it was detectable. I am not familiar with UK petrol ratings, but I would recommend a top quality NO Ethanol fuel. Hopefully, you do not have Ethanol in your fuels; it is bad stuff for small engines and can cause all manner of corrosion and damage, especially in storage if not managed with a fuel treatment. Fuel problems are, by far, the primary cause of engine issues in small engines that do not run a lot. It is also advisable, especially for a genset to be used in emergencies, to remove the fuel once a year and replace it with fresh fuel. Obviously, if you run the unit a lot, that is not necessary.
73, K0ZN