You could check with the support for that particular app to see if they have some method. As far as the native iOS Messages app there is not anything. It requires either a copy and paste into something, printing as you indicated in your post as photos, or a 3rd party program to copy the messages and then send to the printer. I've used TouchCopy from Wide Angle Software and was very happy with it. It works on both the Windows and Mac platform and does a good job. Well worth the price. It does more than just handle messages too. The only thing that may be a drawback is whether or not it can access the information in this app you are using for messages. It may not, you could only try. They have a free version of the software that has limited features, but you may be able to determine if it works with your app.