You can use brass for a backer, copper welds to steel.
Yes, a magnet will jack with DC MIG . You can use a very close arc with it in short arc with gas.
I wouldn't think flux core would work very well. Never tried though.
Look at Eastwood again, they make a tool for forming flush welded lap joints.
You cut a section out, form the lap joint bead around the perimeter of the cut, lay the new cut piece in the formed area and stitch weld it in.
Works great.