In the last week or so I managed to sandblast the lower part of the seat and finish fixing the bad welds. With the paint and rust gone, I was able to notice a couple of little cracks, which I welded. The rest of the seat has only light rust (or none at all), and I can clean up the remaining rust with a sanding disk. Weather permitting, I should be able to paint the seat in the next few days.
I also cut some strips of thin plywood which will replace a couple of tacking strips that were on either side of the seat base.