You are not the only
idiot in the village.
The obvious bike to use is a Surly Pugsley.
You would probably need a trailer to carry supplies.
I'm not sure if its viable, the terrain is often rutted and ruckked so you have to carry/roll your bike over the surface. You also have to deal with snow-melt. If you came prepared to ditch your bike and continue by kayak, you may survive.
This is how real polar explorers do it.
The organised
North Pole bike race do some mechanical preparation of the surface.