You can now control the clipboard access of applications individually in GrapheneOS.
This makes apps only able to see data if the user manually pasted it, with an exception for clipboard items originating from the same app attempting to paste it.
You can go into Settings -> Security and Privacy in Owner profile to control clipboard access globally. You can also go into individual applications in any profile to control that individual app's access. In an individual app, you can go to the app settings -> Clipboard access to control it for that app.
Maybe I'm blind, but I can't find this setting under Settings -> Security and Privacy.
Ah, I just got a grapheneOS update. I see the settings now.