No, you can't save a CR2 file.
You may be able to coordinate changes with both apps to the metadata XMP file - meaning you may be able to tweak the development settings in Camera Raw, press [Done], and have Lightroom see those, but you can't do proper edits inside Photoshop and make Lightroom aware of those. The two apps don't work the same way. Lightroom can only set the development metadata and while Photoshop can do that too, once you open the file into Photoshop proper it changes pixels, adds layers, etc. and the file must be saved as a PSD or other format.
I don't use Lightroom, just Photoshop myself. I hope someone who uses both will comment on their soup to nuts workflow and how they save edited files.
A raw file is just that - the unmodified capture of the camera's sensor data. Once you start editing it, it gets converted to persistent pixels. As Noel said, it may be possible to sync the metadata, but anything else will require the use of a proper pixel format that is able to hold such modifications - whatever they may be - consistently.