The Google chat app has an extra persistent notification - all messages and room are "read" - It is always 1 more than Hangouts. - It does not go away with reinstalls - It does not display in web client or on Android devices
A few months late, but I had the same issue for the last week where the badge wouldn't clear from my iPhone. I finally resolved it by signing out of Google Chat (formerly Hangouts Chat) and signing back in.
I had the same symptoms where uninstalling/reinstalling didn't work, nor were any unread messages in the web app. My iPad didn't show the badge either. Today is the first time I tried signing out and back in on my iPhone. So far, so good.