That fabric is not meant for a washing machine. There are a lot of fabric cleaners that will help it out tremendously without dismantling the whole seat.
If that didn't work, I might be tempted to overhaul the seat...while that's going on (fixing welds, replacing pneumatics & trim, etc) I'd pull the hog rings & get it dry cleaned. It'll turn out better and likely not get destroyed. With either method, I'd e sure to clean a sample first just to be sure. New covers are still available from BMW, but they're about $500 per seat.