So firstly I assume that "LR" is referring to Lightroom-Classic with a Catalog you mention.
"locate BOTH the LR file AND the raw file"
It is a mis-understanding that photos are IN Lightroom-Classic. The "LR file" you refer to is only a reference to the one raw file on the hard-drive. And when you do any Edits in Lightroom-Classic, all the edits are saved IN the Catalog. The raw file is not changed neither is another raw file created.
"In PS I open the PS file, complete with layers, and edit or reoutput."
You should be 'Opening' the raw file to Photoshop FROM Lightroom-Classic [Ctrl+E] and NOT opening from a Finder view. Opening from Finder will add an XMP sidecar to the raw file as it opens through the ACR interface then to a Photoshop document, AND it does not add the new Ps image to LrC.
When Opening from Lightroom-Classic (LrC acts as your ACR) the edits are stored IN the Catalog and no XMPs are created, and when [Save] in Ps the new image is automatically added back to the LrC Catalog library.
" If I want to share the edited file with another person or system, I'm not sure how that would work without my LR catalog."
The only "edited file" you will have is the new Photoshop TIF/PSD. You can 'share' it from its folder location in Finder (Copy and Paste to wherever.),
OR if you view the new Photoshop TIF/PSD in the LrC catalog, you can EXPORT a derivative file in various formats- JPG, TIF, or 'Original TIF/PSD', etc.
CLASSIC to PHOTOSHOP
Moving Between Lightroom and Photoshop - YouTube
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