I would never work in sRGB.
I changed the working color space in Photoshop to ProPhoto RGB.
In Bridge I now get a "not synchronized". If I click "synchronize" Adobe switches the working color space of Photoshop back to sRGB.
And I switch it back to ProPhoto RGB and then I get again "not synchronized".
So Bridge cannot handle ProPhoto RGB?
As far as I know Bridge can handle anything Photoshop can. I think I understand where the problem is:
First set your Color Settings profile in Photoshop with ProPhoto and save it with a name, e.g. MyProPhotoSettings. That is very important. If you do not save, then hitting Apply in Bridge Color Settings to sync will revert back to North America General Purpose 2, in Bridge and in Photoshop as well.
(Note: Just because your Color Settings in Photoshop are set to sRGB, does not mean the document will be opened in sRGB. If your document is ProPhoto, Photoshop will open in ProPhoto regardless of what is in the the Color Setting panel. If you have Profile Mismatching Ask When Opening checked, a good practice, then it would ask if you want to convert the document or keep it as is. Otherwise it will simply open it in the colorspace the document is in.)
Once you save the parameters and choose it as your Color Settings in Photoshop, go back to Bridge and apply sync. Now both would have chosen MyProPhotoSettings. And your images should look identical in Bridge as well as Photoshop.