Cutting door w/ chipboard center
I'd suggest you chuck the door. Chipboard won't hold hinge screws well, nor will it make an attractive surface for the strike side.

If you were desperate to use this door, you could cut a channel in the chipboard edge and insert/glue a 2" piece of solid wood. Getting a really clean surface inside the channel would be hard.

At 31-1/4, you're pretty close to a 32" door, so you shouldn't have a hard time finding something appropriate. 

For what it's worth, many in the door industry call this kind of door "solid core". I'd give volunteers at the Restore a little leeway for getting the description wrong.