I think that it's really one of those "suck it & see" jobs. The problem is there are WAY too many variables - & even working out sensibly what should work, does not mean it will work.
I'd start with clamping the welder to see what it pulls off the mains (& note what voltage your mains is at).
Does your welder manual state the input voltage & frequency range? I.e. how "tolerant" it is? Given how much variability there is in even Mains supply these days (plus wanting kit to function in several different markets) - most stuff is pretty tolerant. If so, you can "cheat" a bit by winding your genny up off-load to (say) 54Hz (so a 3000rpm at 50Hz set is running at 3240rpm off-load to give you a tad more "wiggle room" & that bit more "oomph"). Note that peak hp of lots of small engines is at 3600rpm (60Hz) or even higher (so a 13HP GX390 isn't giving 13hp at 3000rpm, probably closer to 10-11hp.... which is why the 6kva-ish rating).
Weight & portability (& Money). A petrol set is way lighter, & way cheaper than its quality equivalent in diesel. That makes it easier to hump about, while the cost saving pays for a LOT of Petrol. Then folks like Stephill will fit an LPG Conversion (dual fuel) for a few hundred quid - so you CAN get diesel running costs for less money.
Esp if not running lots of hours - then a petrol makes sense, & the weight saving is a biggie if transporting it. Heavier petrol sets can be had in a 2-wheel "Barrow" type frame to make moving one about possible single-handed on hard surfaces (though loading / unloading still challenging).
When you factor all that in.... its obvious that IF a 1-cyl Petrol CAN be got by with - then it ticks every box.
If not, then a Honda GX610/620 V-twin will give you 10KVA, 8kw absolute overload max cheaper than just about anything else of reasonable quality..... but its still a LOT of money, and still a hernia-inducing lump (which would enjoy a drink of unleaded as well!).
You're in the dilemma I used to see people in all the time with mobile equipment - mobile caterers tended to be the stuff of nightmares - needing big power, with nowhere to put it, no way to handle it, not enough weight-carrying capacity etc etc.
If (IF) your measurements hinted that you might get away with a 1-cyl GX390 set - it'd probably be money well spent to drop £100 or whatever on a Hire Set from a Tool Hire place & actually test it - you'd lise a lot more if you splashed out on a set that didn't work!
(I used to see a chap who ran some kind of plastering machine off a Honda Petrol... on paper there was no way, but if the genny was tip-top and run at slightly higher rpms, voltage & frequency than you normally would - pushing the envelope, and I'd bung a new capacitor in it whenever it came in - it manfully stood up to being battered flat-out on a regular basis. I warned him it was damned marginal, to change the oil at least every 100 hours, and to bring it in for a tweak and a new air filter to give it every chance - and he used that setup for a long time).
(That's a very long-winded way of saying you need to do your homework - see if you've any hope, then actually try it & see if you get lucky
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