I am on Android 14 (Pixel 7), and have no trouble disabling the persistent indicator. I make heavy use of clipboard sharing, so it was a useful if bulky thing for me in previous Android versions.
14 allows me a customizable pull-down and I put one for clipboard sharing there. Then, I just long-pressed the persistent notification to bring up some options:
and either from there, or the app’s info page I get the full set of notification options
My app version is 1.30.1
I wonder if your phone has a skinned or otherwise modded version of Android? Maybe they have hidden some things, or have them in an odd place?