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This is all perfectly normal and expected when you work in ProPhoto RGB, and view the file in an application without color management.
This has nothing to do with Lightroom or PE, which work as they should - it's because you are viewing the file in an outside application that doesn't have any color management.
Without color management, files need to be sRGB to display (roughly) right. Set "External Editing" preferences in Lightroom to sRGB (not ProPhoto) - or convert to sRGB in Elements.
Thanks for your very helpful answer.Should the Lightroom settings be set to ProPhotoRGB during the editing process in LR and then switched to sRGB at the time of transferring the file to Photoshop or should it be consistently set to sRGB while editing in LR?
That's not how it works - Lightroom has its own internal working color space and you can't change that (nor is there any reason to).
This is about how Lightroom exports the rendered file to an outside application, and whether that application has color management or not. Photoshop Elements does, so it will handle everything correctly no matter what color space you export in (or "Edit In..." which is basically an export).
You set this separately for Edit In (under Preferences), and for Export (in the Export dialog):
Now - the complications start when the file is viewed in an application without color management (such as Windows Photos and some other file viewers and web browsers). These applications don't read the ProPhoto profile.
Without color management, the file has to be in the sRGB color space to display correctly. Whether you do that in Lightroom or Photoshop Elements isn't important, but it must be done before opening into an outside, not color managed application.
I suggest you just set Lightroom External Editing to sRGB to avoid any problems until you know more about how this works.