Can't export AI file to PSD with layers

How is the final packaging design going to be output/printed? Is it going to be printed from a Photoshop file? Does it have to be printed from a Photoshop file?

Certain types of graphic elements work far better as vector-based objects than pixel-based elements in a Photoshop PSD layout. Logos, lettering and other hard edged graphical elements will print infinitely sharp if they are maintained as vector-based elements. It's possible to use elements from a layered Photoshop file within Adobe Illustrator or Adobe InDesign. Pixel-based photographs and other images from Photoshop can be mixed with vector-based logos, text blocks and other objects.

It is possible to paste objects from Illustrator into Photoshop. You can paste vector-based elements as smart objects. Or you can paste raw AICB paths into Photoshop. They will show up in Photoshop's paths palette as a "work path." Double-click on it to save the path as a named path so it doesn't get over-written by another work path later. AICB paths can be turned into selections to create fills or do other things. Illustrator has been able to paste vector paths into Photoshop before Photoshop supported layers (this is going back to the early 1990's, back when RAM cost $40 per megabyte).

Whenever I have to do multiple passes of copying/pasting elements from Illustrator into a Photoshop layout I'll usually do so by organzing the elements on an artboard or in a box the same size as the target Photoshop layout to keep registration intact. The Paths palette in Photoshop can hold a bunch of different path groups in their own boxed entries. The Free Transform tool in Photoshop can scale the Illustrator objects to snap to the edges of the Photoshop layout.

Can you please tell us exactly how you are exporting it?

It might be a better idea to draw in Photoshop and then re-import that drawing without a background into the Illustrator packaging file.