Can't load "Black Ink - CMYK profile" in the color converter

Some people suggest converting to CMYK Black as a more accurate alternative to using the Dot Gain % profiles.

They suggest doing this in Photoshop by loading the black ink of the CMYK profile required by the printer (for example: "Black Ink - FOGRA XY"). But is this kind of conversion possible in Acrobat's color converter (Print Production > Convert Color)?

This is what I tried: in Photoshop, I went to Color Settings and I loaded the Black Ink (Load Gray), I saved it as ICC (Save Gray) and I installed it. But it does not appear in Acrobat's color converter. Is there a way to load it there?

I found another way to do this conversion in Acrobat, that is: Print Production > Edit Object > Select All > Properties > Convert. It works, but it would be better if I could do this in Print Production > Convert Color. Its tidier to work that way and it gives more control.

I can get the same results by converting to CMYK with these options turned on: Preserve Black and Promote Gray to CMYK Black. But after the conversion, the content will not be identified as Gray anymore, but as CMYK, even though it will use the black plate only (C=0, M=0, Y=0). This might create confusion in the future, so it might not be a good idea.

Thanks