First thing I'd try is plugging the Lincoln into the dryer outlet. Start with a low heat and see how far you can get without tripping the breaker. I've found that I rarely get over 100 amps in the welding I do, even though the box goes to 230.
Robert is right - wire up the big welder to your existing 30A dryer circuit and just don't crank it up all the way.
If you want to see how far you can push it, get an inexpensive clamp ammeter and split the line cord to see how much input 240V power you are drawing at various welder output settings When you get to about
30A input current, mark the dial there as "Maximum for Intermittent Use" and the next notch down as "Maximum for Continuous use".
(That is, if the Lincoln 300 is rated for continuous duty...) ;-)
Why can't you add a bigger dedicated circuit for the welder, or bump up the feed for the Main?
I just got done bumping a Studio Condo up from a 40A main breaker (#8 TW) to a 70A (#6 THHN, 100A meter socket in the stack, 100A rated panel in the Condo) so the new owner could install a new Mini-Split AC to replace the noisy old window unit - Would have gone for 90A to leave more cushion (Bigger is always better) ;-) but they only had
3/4" EMT for the feeder, and we couldn't get #4 wire into the pipe. Legally or physically.
As others have said, hook up your Lincoln. It will draw more than 30 amps only when set to weld at around 150 amps or higher. Unless you're welding a bridge or battleship, you won't need more than that.