Fit a 16A plug to the welder and a 16A socket to the extension and make up a small extension (about 1 foot) for when you don't need the long one.
Replacing the 13A plug on the welder with a 16A one eliminates an unnecessary fuse and the greater resistance of a typical 13A plug/socket combination.
Changing to 16A would be a nice to do but not critical.
For me unless your 16A socket comes from a supply with a circuit breaker and RCD protection then I'd leave the 13A plug on for the little bit of protection the fuse gives. Its easy to damage the lead on a welder and make something in the area live or touch the live wires.
13A fuses aren't life savers but theres a chance it'll blow
Just to clarify my post above (#3) you'll still have a 13A fuse in the plug on the extension, this will protect the extension itself and the mains lead on the welder.