Saving file in PS (not LR)only until project is completed?

Just to be clear: you never save "in Lightroom" or "in Photoshop". The saved file is not inside the application. You save to disk, whatever application you save from. Just pay attention to where you save the file so that you can find it again!

What's special about Lightroom is that it uses a catalog to reference the files on disk. This means that if the file isn't added to the catalog, you won't see it in Lightroom. But it's still there on disk!

If you send a file from Lightroom to Photoshop, a background script keeps track of the file and adds it to the Lightroom catalog when it is saved out from Photoshop. For this to work, both applications need to remain open throughout the process. This doesn't mean the file is "in" Lightroom. It just means the catalog has been updated with a pointer to the file's location on disk.

If you're working on a file in Photoshop, you should always save it as soon as, and as often as, possible. If for some reason you don't want that to appear in Lightroom, just close Lightroom before you save. Then it won't be added to the catalog. Or remove it from the Lr catalog afterwards.

Although you didn't state you desired to do this, I re-edit .tiffs from Lightroom pretty often.  Assuming this is Lightroom Classic and not cloud?

For the files Photoshop send back to Lightroom, I just right-click and select Edit in Photoshop from Lightroom and it gives several options including edit the original file.  It then opens again in Photoshop with all of the layers and such present.

I can then save and close both programs, and re-edit .tiffs as needed.  I essentially use Lightroom as organization for those files.

Just wanted to throw a possible solution out there!

What you can do is not use 'Edit in Photoshop', but use 'Export' and open the exported file in Photoshop. Then, when the project is finished, you can import the result in Lightroom again. Personally I would not mind that the internediate file is shown in Lightroom however.