The difficulty of diagnosing H12 errors is really challenging. One thing I can recommend is using the http-request-id labs feature: https://devcenter.heroku.com/articles/http-request-id With this enabled and some code in your app, you can correlate your app's request logs directly against the router logs and trace what happened with any particular H12.
I'd be happy to help you do this if you're game. Contact me via adam at heroku dot com.
Could you also email me some links to your support tickets so I can check out what happened there?