I have fixed this myself on both seats.
One seat had a really poor attempt at a fix by the previous owner - the original wiring had been cut and fixed with a solder job that was not supported or stress relieved - pretty shoddy work in an airbag circuit!!!
I won't go into how I fixed that, this site may not be comfortable with people encouraging other people to play with the actual wiring of safety circuits. The actual wiring is not a problem (unless an incompetent ape has been working on your car before you bought it) - it is the connectors usually.
What I can recommend is stress relieving the connectors (remove battery negative cable before disconnecting any of the connectors, don't replace it until all airbag etc connectors are connected correctly).
I've seen the diagrams from the official BMW fix and it involves taping the cables and connectors in an S shape - ie start off with the connector turned the "wrong way" and make the shape below, then wrap it tight with electrical tape.
*Sorry, this looks right when I type it but when I submit it, everything zaps to the left*
*you'll have to imagine that the \ | /above the connector are about two inches right of where they appear*
Don't pull cables so tight that they have any very sharp bends in them (bad for wires)
Don't make the bends too relaxed - or you'll lose too much cable in the S shape and may run out of cable if you push the seat all the way back etc. The BMW fix involves putting in longer wiring to allow extra for this looping around. You'll just have to be careful!
Check your new setup by running the seat back and forth before you reconnect the battery - make sure nothing catches or gets stretched.
Doing this should reduce stress in the connector and prevent those split second resistance changes that trigger the fault light.
This will not reset the fault light, you will need the services of your local mechanic-with-a-laptop to reset this. Once he's done it, and is still standing nearby - you can try being cheeky and whizzing the seat back and forth a bit to check the quality of the fix. If you wait til the next day to try this, and the fix hasn't worked you may end up being charged again when you go back to have another fault cleared.
This worked for me but I fix electronic items all the time - if you're not confident don't do this!