Its possible, But you'll have to do a good job fitting the wood piece in there.
I think that simply taking a rectangle chunk out of the neck through end, going through the entire thickness of the bass is probably not such a good idea, since the two surfaces that go alongside the neck will probably be subjected to "crawling" due to string tension, and the remaining surface would be end-grain to end grain contact (weak).
I would suggest increasing the gluing surface with some sort of a "step" like a rabbet joint, or simply not to go all the way through in order to provide a gluing surface thats flat to the direction of the string pull.
Also, you will not have a bridge pickup since you are moving the bridge away from the pickups, and routing the piece of wood you are adding would severely weaken it.
hope it helps.
B