On my iPad, I added an email and password to my Overcast account. On my iPhone, I disabled iCloud storage for Overcast. Then I could use the app like normal and all my data immediately synced.
I’m not marking this as resolved, because it’s an iCloud problem and not an Overcast problem specifically. It remains frustrating to me that iCloud is a huge black box, and I prefer not to use it unless I can see the synced files in my iCloud Drive directory on my Mac and thus back them up via Time Machine. /rant