No, unfortunately, there is no way to use an Android device to ‘cast’ to a Sonos Beam.
To clarify, there are several music service apps that allow you to cast audio to Sonos. This is basically telling the service to stream directly to Sonos rather than to your phone. In that regard, your android phone isn’t doing the casting. Amazon, Pandora, Spotify, are some of the services that can do this. I don’t know if Podcast Repulbic is one of them, but it appear not from your testing.
Thanks Bruce. It's disappointing that casting is not possible.
But is there a way to play my podcasts from Podcast Republic via the Sonos app?
https://support.sonos.com/en-us/article/add-an-internet-radio-station-to-sonos
If there is a radio URL, then you can. However, it sounds like you’re looking for an on demand podcast rather than a streaming radio service.
Another option to look into is whether you can chromecast to your TV, or device connected to your TV. For example, I have an Nvidia Shield streaming box, so I would be able to chromecast to the Shield, which would then play the audio on the TV and my connected Sonos Arc.