Unfortunately, when you are streaming directly from your console you do not have the option to implement a browser source in your stream. The Sound Alerts browser source is required to use Sound Alerts on your stream. However, you have the option to use one of Elgato's capture devices to capture your gameplay to your PC or Laptop and stream with any streaming software from your PC. By doing this, you have way more options and freedom for streaming, using plugins, addons, and more.
If you do not own a PC there you can alternatively use one of the Cloud Streaming Software such as Lightstream.
Using Elgato Capture Devices
Elgato's website and devices can be found here
Compatible with most streaming software are the Elgato hd60s, hd60s+, hd60 Pro, and 4K60 Pro.
For the correct setup, please use this video guide from Elgato.
Once you connected all the cables correctly as shown in the Elgato description you get with the Item, you can set up everything in your OBS Studio setup.
Right-click your sources tab, press Add and select "Video capture device".
Select your desired capture card to "Device".
Right-click your sources tab, press Add and select “Video capture device”.
Select your desired capture card to “Device”.
Press the "Add Source" button and click on "Devices (webcam, capture cards...) > Video > Game Capture XX
Using LightStream as your Cloud Streaming Software
If you don't know how to set up LightStream, please follow their video guide here
To add the Sound Alerts browser source to your LightStream setup, head over to your Sound Alerts dasboard and copy your browser source URL.
Now click the "Add layer" button in LightStream and choose "3rd Party Integrations" then select browser source.
Paste the copied URL here and play a test alert to test your setup.