Your machine has a 12ga cord, 6 feet long. (I think) You are asking if exchanging to a thicker 10ga, but longer (15 foot) cord for portability reasons could be detrimental to performance.
Here is an analysis from an internet
voltage drop calculator showing both cord configurations that you mentioned. Voltage at the end goes from 119.6v to 119.4v with the 15 foot cord, so miniscule it probably couldn't even be measured in real usage. If a 15ft cord makes your tool more useful, for a .2v difference, just do it.
IMHO, changing the cord is extra trouble that you couldn't possibly notice the benefits from. I have a 15-foot 12ga "RV cord" that I use often and have never noticed the difference (between "ext cord", & "direct"). I weld often with generators that are slightly undersized, and often need all the power I can get.
Using the ext cord (or not) is not noticeable. And this is extended, in ADDITION to the 6 foot cord on the machine. A 12ga "RV cord" is a pretty hefty cord though it's about 1/2" diameter. I have not tried welding at full power with garden-variety extension cords but have often welded thinner stuff (like 1/8" etc) with garden-variety ext cords.
I don't think anybody can notice the difference in fractions of 1%. You certainly won't damage anything.
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EDIT: The 3rd part of the (new) pic shows the effect of added 12ga ext cord 15feet long for a total of 21 feet. The voltage drops from 120v to 118.67 (minus 1.33v) when welding at a full 20A, which is ONLY when you have the machine set for FULL POWER.