You might be able to strap the bike down to a 'piano dolly', the type used in moving big furniture.
It would be easier to manuever. Usually all four wheels swivel. Maybe take the front wheel and fender off of the bike to shorten it up a bit..
Harbor Freight has them, as well as other places.
You'd want to make sure the bike is off the ground a few inches,maybe build up the wooden bed a little, and then put some means to strap the bike down, like you'd do if you're using a bike jack.
If you can find one for $20-40, it's probably worth it.