I ride mountain bikes, but not full suspension, so take what I say with a grain of salt. I think it is probably possible to replace the shock with some type of rigid bar, but I don't know how stable that old rear triangle will be to want to actually place a human being in some plastic carrier and ride.

We have 3 kids so I can give you this advice on this. Rear bike carriers are a pain and the extra weight throws the balance off. Years ago we decided that a bike carrier was not the way to go and we went out and spent the money on a trailer. It was a Cannondale Bugger and it was bulletproof. Pulled the kids all over the place when we camped. You can't buy one anymore, but there is probably something out there that will work just as well. Don't buy a bike, buy a trailer.

John