FWIW as someone who has an off-grid property, and a few Lister, and who does quite a bit of welding . . . this is my opinion
(and as others have commented) the load of striking an arc, the possibility of damage, the lack of portability with the Lister, the value of the gen head you may burn out - taking all these things into consideration, about six years ago I paid $450 for a 9kVa Chinese petrol generator. It runs the welder easily, it's portable-ish. It is unfailingly reliable (although it does have an EPA-complicated carb design - thanks Uncle Sam) and it costs nothing to run for an hour here or there
I reckon that's a safer and better option. The other day I loaded up the Markon head on the Lister (doing something unrelated to this) and, listening to the engine note change and the revs "droop" a bit, I felt I was doing both the Lister and the Markon a dis-service
I have played with those little Chinese Honda-clone "Chonda" generators almost daily in my work for the last decade (I own six or eight personally, and we have another six or seven in the depot here) and I have a lot of respect for their cheapness and their ability to do a job. Hand on heart I can say I have never seen one "blow" up - although they will if abused, of course
They are much-derided as "Screamers" in some groups - but really they're just a machine to do a job. If you understand them and are comfortable with basic maintenance - then they are a good option. It's easy to move one around the place in the boot of the station wagon and, when it's where I want it, I park the wagon in front of it so it cuts out a chunk of the noise
You do need about 7/8000 watts for the welder to run effortlessly and for the generator to have a long life
Just my $0.02