Interesting. I had an app, that had pretty much this functionality. Read the SMS from a phone and send a push notification with the content to a second phone. Google killed it though, because they said it does not have a legitimate reason to read SMS. Still a bit bitter about it.
You can rig this up yourself with Tasker and Pushover. There is an plug in for Tasker that enables using REST API calls and with variable names in the call so you can capture the text of an SMS. I used to do this for an old hotline phone.
