Does turning on the bathroom exhaust fan help change the noise at all? It might be that having airflow in that vent changes the sound.
You could try an actual white noise machine. A device like this, for example:
You can adjust it to make more or less white noise. It does a great job at drowning out other sounds. If you want to get an idea of how well it works, you can try using a regular fan on high. The whooshing of the air will be similar to the white noise machine, but the dedicated device will do it better and make a more full white noise.