Get "estimated reply time" from LSW Twitter cases and display on external website

How do we display "estimated reply time" from LSW for Twitter cases on our external website?

Background: We are currently displaying an "Estimated reply time" for our Twitter channel on our help website. This is to encourage users to contact us via Twitter since the reply time is quick. However, the reply time varies from 1 hour to 5+ hours. We want to have this estimated reply time update automatically from LSW.

Any suggestions on how we can do this? Is there an API for this that we can call from an external website?