Bike in Punta Arenas
The only contact I have for this is Gonzalo Yutronic, Carrera 666, opposite the cemetary, but I expect he popped up in your search.
Realistically, your best bet is to buy a bike and sell it when your trip's over, hopefully from/to another traveller. Post in the Travel Bikes for Sale or Wanted forum - you may get lucky! Rental will always be an expensive option.
Regards and good luck, Mick