I think the two clamp method would probably work, but there is at least a slight chance that it could crack the body wood if you are not careful. No chance of that with veneer.

...but speaking of body wood. The pics make it clear that it's an open grain wood. So not basswood, alder, maple or poplar. It looks like ash to me because it's light and has pronounced grain. Good that you ordered some filler ;-) You wont' need it on the neck, but you will on the body.