One LR setting to check: go to Catalog Settings, Metadata tab, 2nd checkbox down:
"Include Develop settings in metadata inside JPG, TIFF, PNG, and PSD files"
If you've successfully written LR metadata to the PSD file, then its modification date and time updates to reflect this - as seen in Finder / Explorer.
Another setting to check: Bridge can either maintain its own metadata database, or else it can refer dynamically to the metadata saved on the image files. Perhaps if set to the former, it may ignore settings subsequently changed by LR externally.
It's none of my business - but I am always curious on why Bridge might continue in play - as a front end - for someone who is also using Lightroom...? Doing that seems to me like both a complication, and a restriction... but there may still be sufficiently strong reasons anyway. That would be interesting to know about.
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Files are not changed or saved in LR. All edits done in LR, whether to a RAW file or any other image file type, JPG, TIFF, PSD or PNG, are stored in the LR catalog file and Applied to the image when displayed in LR.
With proprietary RAW files those edits can also be written to a XMP file and with other file types they are saved to, written to, a special place inside the file itself. But the original pixels of the image file are never changed.
With RAW files and the edits written to a XMP file Bridge should be able to display them. With all other file types I'm not sure Bridge has the ability to read the special edit data and display them.
You will need to send that PSD with the new edits to PS and save it as a new file for those edits to show up in Bridge. You will get a dialog box asking to send the original, a Copy or a copy with LR adjustments. you would select a copy with LR adjustments then Save in PS.
What I suggest is you don't use LR to further edit a PSD file and use the Filter system in PS to use the ACR filter to make those new edits. Then of course save it in PS.