How to add Paste button in UIMenu such that the system "allow app to paste" prompt does not appear

There's no way to get around the prompt when using the pasteboard programmatically. This is a privacy protection feature Apple has implemented that you cannot opt out of to protect users from apps that would like to data mine or fingerprint them by grabbing pasteboard contents in the background.

The only way to get around it is to use something like the UIPasteControl, which must use a different internal pasteboard API that does not trigger it (because the system knows that it was from a UI interaction).

Because of that, you may be able to use something like the SwiftUI version in a SwiftUI menu: PasteButton. SwiftUI is sometimes smarter about that, and because of the aforementioned reason the PasteButton API will (should) not trigger a permission prompt.