I have no idea if this first option would even work as I can find no exclusion for a process in the web shield, only exclusions for specific sites.
One of the problems for many streaming sites is that they do so over HTTP ports and don't stick strictly to HTTP protocols and that is what the web shield expects over those ports that it is redirecting into its localhost proxy to scan the content.
So I don't know if this is an issue with a win8 app or the site streaming the Radio.
Does the app run independently of the browser, e.g. in its own window not your browser ?
If so you can have avast's web shield only 'Scan traffic from well-known browser processes only' in the web shield expert settings Main settings window. That however, would also exclude other apps/programs other than browsers connecting to the internet, so could increase the risk.
You could use the avastUI, Settings, Troubleshooting, Redirect Settings, WEB and enter the IP address for the streaming site in the Ignored addresses: field. This effectively that it doesn't redirect that IP through the web shield proxy, where as adding the domain to the web shield exclusions (as suggested in the link you gave) still gets redirected through the proxy I believe, but doesn't get scanned. So this may not have the desired effect, e.g. no interaction with the web shield proxy.
My suggested option would be setting the IP address in the Ignored addresses field.
