Alternatively, Is it possible to “cast” podcasts from my podcast Republic app on Android phone to my Sonos Beam?

The cast icon/button appears in the Podcast Republic app, but my Sonos Beam doesn't appear as am option

No, unfortunately, there is no way to use an Android device to ‘cast’ to a Sonos Beam. 

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?

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.

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.

Thanks Danny. I have a subscription to Amazon prime so I can steam music to Sonos beam but I mostly listen to podcasts so it's a bit of a dud that it won't behave like a regular Bluetooth speaker. Or TV is in another room and because it's an old plasma (which still has an excellent picture so we haven't upgraded) the Sonos beam won't connect to it. So I'd hoped I could use it as a regular Bluetooth speaker. Oh well, pretty much useless until we upgrade tele ‍♂️

A side note to your post. The Beam can connect to non-HDMI-ARC TV’s using the optical converter that came with it. Then the Shield/Chromecast route could work.