ICC profile for web

In the end you need an sRGB image for the web. You can edit in Adobe RGB if you want (to work in a bigger color space) as long as you convert to sRGB when saving as a JPEG, which Save for Web and Export As do automatically (but Save As does NOT).

Firstly, you mention "Google shopping images. We found out that PSD defaults to - - - "

- do not place PSD files on the web, generally for images jpeg is the most used format. Maybe you're doing that already I don't fully follow your description.

Secondly - you mention "convert to working RGB" - that RGB could be any colourspace as it's set in Photoshop's "color settings" 

For internet use your best chance for reasonable reproduction is to convert images to sRGB AND embed the ICC profile when saving the web version. 

If the file will ever be printed, archive a copy in the original colourspace and make a copy for conversion to sRGB.

You mention the camera jpeg file, best to process the raw properly and resize before making the Jpeg, a Jpeg image file does not survive resizing well. 

I hope this helps
neil barstow, colourmanagement net :: adobe forum volunteer:: co-author: 'getting colour right'
google me "neil barstow colourmanagement" for lots of free articles on colour management

What is the actual process? 

Using the setting »Convert to Working CMYK« is generally a terrible idea.

»Preserve Embedded Profiles« makes sense in most situations. 

Edit: … in my opinion, that is. 

What is the images’ original Color Space? 

If it is Adobe RGB or eciRGB v2 etc. you can work in those Spaces, save the layered psd in those spaces and convert to sRGB on exporting.