Until you respond to the request for ACTUAL make and model #'s for your welder AND generator(s), all the answers you get are only going to be WILD SPECULATION at best - preferably links if possible -
1 - virtually ALL generators have two "ratings" - one is the PEAK wattage (mainly a number for liars to sell you less than you need) The second is CONTINUOUS wattage _ OK for a CONTINUOUS load, but STILL not good enough for a SUDDEN load like a welder needs when striking an arc - for that the generator should have at least half again the wattage capability.
2 - There are very few (if any) companies who sell INVERTER welders who will honor their warranty if you use a generator that is NOT "clean power" - this means less than 5% THD (stands for Total Harmonic Distortion) - That specification is like any other spec; if they do NOT brag about it, they ain't GOT it - the least expensive clean power genny's I've seen that DO have it, are some of the Power Horse branded genny's that Northern tool sells - those tend to start pricing close to $1000, so not a cheap "bandaid" either.
3 - If you don't care about warranty, you MIGHT get away with using either of the genny's you have to run your inverter welder, and YES - you WILL see enough difference on 240 volts to run at least one size larger rods. No guarantee just how LONG you'll get away with it.
Sorry if I'm bursting your bubble, but like everything ELSE in the world, "TANSTAAFL" (acronym from an ancient SciFi book, means "There Aint No Such Thing As A Free Lunch" ... Steve