This can come down to device hardware compatibility. If the app requires permissions based on hardware that the device doesn't have or support then Play will prevent the installation. This is more of a consumer protection mechanism in this case as its designed to prevent poor end user experience in the Public Play world. In the enterprise / Line of Business world however this can be a problem. The TC56 and TC57 devices both have cellular radios whereas the MC9300 does not so my guess would be the Phone permission is what is causing the conflict. 

Interesting that it would have worked on a demo MC9300 however. Perhaps the app was installed through a different mechanism than Managed Play for the demo? Or maybe there are new permissions being requested by a slightly newer version of the TIMS Assistant app that are no longer compatible with the MC9300.

EDIT

Actually, I bet it's the Camera. There are different PNs of the MC9300 that have Cameras, and less premium versions that don't I bet you demo'd with a version with the camera but your current MC9300s don't have a camera so they're running into this permissions roadblock:

Is this something I would get with the application author in order to have corrected? These applications are built for us so I am thinking it could be something on their end that might need to be changed.

We found out that the model used in the demo was the exact same, however it was installed using an APK and not from the Google Play Store. When we got it installed using the APK, it now shows installed from the Google Play Store but I am concerned whether updates in the future will install. He then said he was going to make a "binding" but I don't know what that is.

If the developer can provide you with an APK then just package that into a SOTI Package and install the app directly on the devices via that mechanism. The app should run even without the camera permission if the developer has coded it to do so. This will bypass the built in protection mechanism in Play that was originally meant for consumers anyway. 

When Play recognizes and app that installed on a device matches an app available in the Play Store, it will automatically upgrade that app for you whether you want it to or not.  You can see this in action with the SOTI AE agent that will be automatically upgraded by Play even if you used a different mechanism like StageNow to install it in the first place, and even if you haven't explicitly approved that app in Managed Play.  Therefore, I think the app will continue to receive upgrades from the Play Store automatically on your device after you install the APK via Package, but I'm not entirely sure because it may still fail that Permissions check.