It depends on whether or not you need to charge batteries while powering loads. I have a 3000VA Multiplus and I used to get by with a Honda EU2200 but I only expect to charge batteries and run minimal loads if I have to drag out the generator.
I now have 2200 watts of solar panels and I have not had to use the generator since I added the last few panels. As I add more loads I'll add more solar.
It depends on whether or not you need to charge batteries while powering loads. I have a 3000VA Multiplus and I used to get by with a Honda EU2200 but I only expect to charge batteries ...
How low can the Multiplus input amps get limited? Might well get by with a 1000va Honda if looking for the minimum.
Keep the load at 80% or less of the generator rating for best results.
I have a $450 Wen DF475T (dual fuel, 3,800 watts on propane), and a $400 Chargeverter. I charge the battery directly.
Chargeverter lets you use whatever cheap generator you have.