Doesn't make sense to protect a 20 amp piece of equipment with a 40 amp breaker.
This is not a concern. Circuit breakers don't protect the machine. Circuit breakers protect the building wiring that feeds the machine. The machine itself has it's own overload protection/breakers/etc built in...your breaker does not need to protect it.
Always size breaker to the wire in the home and to your local electrical code. Don't rely on a Lincoln Manual for this. Note when you Google search "super lag breaker" the only people asking about it are people who are reading their Lincoln Welder manuals....this aughtta tell you something about the quality of the advice in that manual. Ignore what it says and consult electrical code.
You stated you're using 10 gauge wire, which is rated to handle 30 amps continuous. For that, a 30 amp breaker is the standard in most any electrical code. If you have a 20 amp welder, then the 30 amp breaker won't ever blow, so it's appropriately sized.
My suggestion for household wiring: 30 amp breaker, 10 gauge wire. It's safe, has adequate capacity for the 20 amp machine, and should meet your electrical code. Get a plug that handles 30 amps on the wall (meets code) or greater (which will probably violate code...but will be safe for YOU unless/until someone else 40 years after you're dead decides to plug in a 50 amp appliance in this location after replacing the 30 amp circuit breaker with a bigger one but not the 10 gauge wiring...thus the code violation. Codes somehow try to predict the actions of stupid people of the future, while at the same time instructing the stupid people of today).
Maybe this is clear as mud? Anyway...that's how I'd do it...and frankly I'd skip the whole extension cord plan and just replace the entire electrical cord on the welder with a 12/3 flexible (or upsize to 10/3 no harm) up to about 50 feet...beyond that you have voltage drop issues. First thing I do to every welder and major portable garage appliance I ever buy...dump the 6 foot factory cord (which you will probably notice is a 14 gauge FYI) and replace it with a nice 40 footer 12/3, straight from where it plugs into the power switch inside the unit. I know you're worried about warranty in doing something like this, but that's not something I ever worry about so you can make your own call as you feel comfortable. 220V extension cords are likely a residential code violation in themselves in most jurisdictions...usually not recommended but I agree entirely safe if properly constructed and commonly used many places.
