A package can help you to keep the device's screen on programmatically without any permission.
Add this to your pubspec.yaml (or create it):
dependencies:
flutter_keepscreenon: any
Then run the flutter tooling:
flutter packages get
try {
await FlutterKeepscreenon.keepScreenOn(true);
} on PlatformException catch (e) {
print(e);
}
Details in example/ folder.