I don't have a 3Dlut plugin but I do a lot of Photoshop <-> CameraRaw editing.
what I often have done is very similar to your workflow but when I put my image in Photoshop, I have it as (raw) Smart Object Layer which retains full raw editing capacity (no conversion to TIF, all setting is intact.), . Maybe you can try that? I use CameraRaw rather than deal with LightRoom Classic's database things. but I think LightRoom is using the CameraRaw's engine so I guess you can tell Lightroom to open your raw image in Photoshop as a smart object layer just like CameraRaw and go from there. (raw) Smart Object in photoshop can export back to .dng as well. Usually, I will just keep it as PSD. and the original DNG will automatically turn to a backup file.
Note that the raw smart object layer in Photoshop is not the same as converting a layer to a smart object because raw smart objects can export back to DNG as mentioned but typical smart objects or layers can not export back to DNG.
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Or another option in the Photoshop layer (not sure if it will work with your plugin), you can try Filter > CameraRaw filter. That will give you tools similar to Lightroom and hopefully, you can see the plugin there. But the best way I have found is still the same. Import DNG as raw SmartObject into Photoshop layer. That way we have full function DNG raw file which can be export back to .dng again if needs.
this is where to find the import DNG to Photshop smart object. (skip #2)
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Here you go, I found an instruction for Lightroom to do the same.
“– Lightroom : When you have completed the edits in Lightroom, you are able to open the image for further edits by selecting “Develop mode: Photo/Edit/Edit in Adobe Photoshop”, however, if you choose “Develop mode: Photo/Edit/Edit as a Smart Object in Adobe Photoshop”, the file will open inside Photoshop as a Smart Object. When inside Adobe Photoshop you can round trip back to Adobe Camera Raw (Note not Lightroom) by clicking on the image thumbnail inside the originally created layer.”
Source: https://blogs.adobe.com/richardcurtis/2012/06/29/improve-your-camera-raw-workflow-using-smart-objects-in-photoshop/
*Note, I think the author made a small mistake when he said: “Photoshop will keep the link back to the Original RAW file”.
*When say “link”, it usually means; if I move parent file(PSD), I have to keep the structure of the children files (DNG) as is, or the parent (PSD) won’t be able to fully function because It cannot find the children files.
As on my workflow, Photoshop does NOT link back to the original raw file. The raw DNG is instead “embedded” into Photoshop. And we can export it back to a new DNG. The original DNG will remain as a separated file.