If your customer has sent you the completed design, I assume you don't actually want to use the layers in your template (as they may not match the layers in the customer-supplied design). So, is it safe to assume that, really, you only want to copy over the view, logo, notes, and title block across?
Should that be the case, I don't see why you could not use either the Tools -> Create Module command to create a reusable module of those items from your template and then place the module into the customer design. This would have the advantage of allowing you to refresh that module in the future if you needed to make changes to the template objects and push them into any designs that reference that module.
The other option would be to use the File -> Export -> Sub-drawing and File -> Import -> Sub-drawing to make a clip file of the template objects which you could then bring into the customer design. This would be more of a one-time thing, since the intelligence is lost after it is imported, unlike the module (which is why in your place, I would likely go with the module-based flow).
Since I don't know exactly what the contents of things you're copying over consists of, that's as helpful as I can be.