Are you giving the client a pdf for the website? I think you may be overcomplicating this. When you choose your pdf export settings, you can choose RGB. You don't need to make any changes to the original document.
The appearance of CMYK black (0|0|0|100) is controlled by the document's CMYK profile—on a printing press 0|0|0|100 is not absolute black.
If you want to override the profile and force 0|0|0|100 black to view as absolute (0|0|0 RGB), set Preferences>Appearance of Black to Display all Blacks as Rich Black.
If you want your RGB black to imitate the profiled CMYK black make a 0|0|0|100 swatch and convert it to RGB. The default Adobe RGB to SWOP CMYK profiles converts 0|0|0|100 to 39|37|37.
Yes, I'm emailing them a pdf, which the web administrator will add to the website.
I had changed the document color settings from CMYK to RGB. Then I thought I also had to convert the Swatches. As usual, I am guilty of thinking that things are far more complicated than they really are. So I didn't have to do any of that?!
I just now changed the "Transparency Blend Space" to Document RGB, following the instructions from an error message. Then I generated a low-res PDF with Colour Conversion: Convert to Destination (Destination: sRGB IEC61966-2.1). All the blacks now look consistent.