Search for Terminal in Spotlight.

Open Terminal and drag the Shazam file into the Terminal window as seen in the picture.

Press the Return key.

This time when you try to access the microphone, Terminal will ask for permission.

This is because apps built into macOS (not apps downloaded afterwards like in the App Store) don't need SIP to ask for permission to access. So, Terminal can gain permission while Shazam cannot.

Unfortunately with this method, you face two main issues. One, this is not a permanent fix. You will have to do this every time you open the application. So, if you use the application a lot, opening it through Terminal may get old. The other issue is that you cannot close Terminal while the app is running. That would quit the app.