Is it possible to change my whisper sound effect?

I find it rather jarring and grating. It helps snap me out of a stupor, but getting them as normal messages from friends is rather intense.

Is there a way to change this sound effect without doing anything rule breaking?

Nope.

Technically you could disable the sound effect in-game and make a standalone app to watch the Client.txt for the recently logged incoming message and a custom sound effect playback. But this setup has quite high latency and not enjoyable to use at all.