Can I warp a raster image to an irregular shape (easily & accurately)?

Editing the image would require to update the Linked Smart Object in the four Smart Objects, but that could be automated if it is likely to happen often. 

In a case like this I would consider splitting the image into four Smart Objects (containing a linked Smart Object to link to the same separate image) that split the original image into four rectangles. 

Depending on the shape of the four irregular shapes the seams of the transformed SOs may not line up perfectly, though. 

Could you provide the target shape as vector data or at least at a higher resolution? 

Depending on your image, Puppet Warp is another alternative or even Perspective Warp. Then, as advised above, clip it to the shape layer.

Dave

I don't know of a way to do that automatically in PS, but you could use Edit > Transform > Warp and pull it manually into place. That won't match all the edges of the box though, just the overall shape. You could do that first and then make a clipping mask with the box layer to cut out the edges. Not sure if that's the result you want, though.