How to stabilize and add additional power to an existent line?
I was wondering, would it be possible to stabilize the line and
provide the additional peak power say, by placing an accumulator
(battery + electronics) in the middle/in parallel?

In the end, the only practical and viable solution is to rewire the the AC feed cable to where you are operating the welder. Trying to interface a DC battery with a 220 volt AC power line is fraught with difficulty and there's no guarantee that what you might be able to cobble together will achieve what you want.
If your welder runs OK when close to the 220 volt AC distribution board and only fails to achieve full power when being fed along the cable, the thing to fix is the cable. Don't be like the old woman in the song that swallowed a spider to catch the fly (et cetera).