Why won't the iPad Podcasts app download my MP3s?
Since you have access to the server that is supplying the podcasts, you might have to look over the access logs and correlate the time with when your iOS device failed to download an episode.

If for some reason you can’t or won’t look at the server logs, next easiest is to set up a web proxy like Charles proxy - https://www.charlesproxy.com/ and generate more logs. 

Last useful would be downloading Xcode and running instruments on the Mac to connect to your iOS device and look over the system and use console app to attach to the iOS logs as you attempt a download. 

Let us know what you did to figure this out!