Not to be contrary, but I feel like this isn't actually possible. The alerts is just a webpage you use as a source in OBS. The content of our OBS stream doesn't have any effect on how it uploads to the server; it's still gonna be using the bitrate and resolution you selected. You could load any website in OBS, it doesn't matter since Alerts isn't an OBS feature. You can view it as a browser window while you're streaming to see it in it's bare form; it should still be in use even when its not attached to OBS unless you turn it off via the Alerts setting.
The only thing I can think of is your computer isn't capable of adding a webpage source with that much going on. You might want to try adding another sophisticated webpage, like maybe a YouTube video or something, as a source and seeing how your feed handles it.