i have been running a 4000 w generator to power my Hobart AC/DC stick welder now for several years. I posted a very similar question to yours here at that time and lots of people told me it couldn't be done, although they had never tried it. A couple of people had and recommended using thinner rods. So when I'm out on my property welding, or at a friend's house with no welding plug, I run 3/32" 6011 with no problems. Amps range from 70 to 130 depending on material thickness. Any more than that (which isn't needed), and the breaker will trip on the generator after a few minutes. My rationale at the time was to go ahead and buy the welder and try it since I already had the 4000 watt generator. If it didn't work, I could always buy a bigger generator. The combined cost would be way less than a dedicated generator/welder unit, and the generator could be used for a lot of other things. As it turned out, I didn't have to purchase a bigger generator, and I was set up welding for under $300 for the welder and Harbor Freight auto darkening helmet.
Disclaimer - I am not a professional or even close. My experience consists of a large rv/boat barn, a smaller tool/lawnmower barn, and lots of smaller projects and repairs. I am currently building an entrance gate/fence on my property and it has all been built using generator power. The welds aren't beautiful like a mig, but they work and a grinder makes them pretty if you want.