Saving PS edits to raw files to be able to continue editing in ACR / view in bridge

Simply put- No.

ACR is a 'Parametric' editor that can 'save' editing work as metadata (either as XMP sidecar file, or embedded within some file types).

Photoshop is a "Pixel' editor that needs ACR to 'render / interpret' raw images. Once you 'Open' an image from ACR you are now working with rendered RGB pixels and edits can only be 'saved' in the resulting .PSD, .TIF, file that is created, and not as metadata.

So- Different processes. Photoshop edits are not the same as Camera Raw edits. You need to preserve the images edited in Ps as independent derivative files.

The only way to edit the raw image data AFTER the image has been opened in Photoshop is to first open the image from Camera Raw as a smart object. Once the image has been opened into Photoshop it is no longer raw image data. Photoshop, by itself, cannot edit raw data. That is why the camera raw plug-in is needed. However, if you choose the option to open the image as a smart object then you can double-click on the smart object in the layers panel and recall the raw data in Camera Raw (the plug-in) to make further adjustments. But Photoshop is not capable of saving the image as a raw file. It has to be saved in another format. Consequently, all of your raw adjustments have to be made during that initial transition into Photoshop even if you use the smart object option.