The printing company wants to see the dieline & adjust your bleed if necessary because most likely, if you don't have any experience in packaging, your print-ready file might not be right. Printers have some specific requirements that they don't always share with designers because it's too technical and specific to their equipment or process.
The real technique:
For that kind of project, you should assemble everything in Indesign, QuarkXpress or any other publishing software; you import in it your background with the Photoshop effects FLATTENED saved as a eps, tiff or pdf file. You can import as many flattened images as you want. The vectors are usually done in Illustrator and saved as eps, pdf or ai, and imported as well in InDesign, in separate Illustrator files. Not copy/pasted, imported.
All the fonts and linked images are in a separate folder in the root folder where the main Indesign file is and should always follow that file. In Illustrator, it's not a bad idea to vectorize your fonts on the final print-ready files or logos; if you don't, make sure you send them to the printer as well (every single font used in your design, logo, graphics, etc.).
Designers usually send a print-ready PDF, and all this package above IF the printer requests editable files. But most designers (and for good reasons) are resistant to send editable files and prefer to do their revisions themselves, and re-send a new print-ready instead.
The other way
You can always do this with Illustrator using the exact same technique I mentioned above but don't do it the way you explained in your question with all the layers and effects from Photoshop.
Merge some layers, simplify that PSD to the maximum, then import. That will greatly help you fix your issue. Keep the text layers un-merged if possible so they can be converted to vector. Normally, you shouldn't import special effects, layer masks, blending and all these layer effects from Photoshop to Illustrator to avoid losing layers or getting bad conversions.
If you decide to work in Illustrator, I suggest you don't try to import your whole Photoshop file at once but rebuild your file in Illustrator instead. Whatever can be done in vector should be done in vector, especially texts. It's more work this way but it's more accurate too and the printed result will be super sharp! You already know how to import layers in Illustrator, you should be able to save a lot of time and keep most of your texts and logos in vectors.
Please have a look at the first part of my answer on this question to
Software for Calendar Designing
You can also send your editable files to the printer (the layers of the Photoshop parts) but frankly, it's better to ask what are the requirements, revisions and guidelines, and do the changes yourself if you can, and if there's any. If your file is very complex, they might end up asking you to do it anyway. If you do send the editable files, make sure to identify the print-ready and the editable files properly to not confuse the printer.
For the dieline, have a look at this: How should I set up dielines so that they are not printed?
You'll learn to love Illustrator and Indesign, especially Indesign (or
QuarkXpress) for that kind of project. Once you know how to use it,
you'll see how it can save you a lot of time and add way more
precision to your work! You can ask your printer or their prepress department about it and how they prefer to receive files. And how to prepare them.