People have been posting for a while about having issues with re-installing Modern Warfare III on the PS5 after removing it from the Call of Duty app.
I tried many of the fixes that worked for others, and finally figured out what worked for me.
I have the PS4 disk version of the game that came with the digital key for the PS5 update. I have the PS5 version of the COD app,
I decided to download the PS4 version of the COD app and give that try. After it installed, I clicked on MWIII within the COD(PS4) app and finally had the option to install the PS4 or PS5 version of the game, separate from the COD app.
Hope this helps someone who has had the same issues. How frustrating this has been. Please leave a comment if this was helpful to you.