The Lutron SmartBridge has an option to set a static IP. You should use that option. Select an IP address outside the DHCP range of the router. You may need to adjust that range if it take up all of the addresses (2..255). Once you give it a static IP, you will lose connectivity to it from the mobile app for a minute, and the app will need to find it once again. You may need to log out and back in.
Enter that new IP into the Lutron Integration app and hit Done.
I could be wrong but I think he's saying he is updating the IP recorded in the Hubitat's Lutron Integration app, but that it won't save that new setting permanently ... over time the IP configured in the Hubitat Lutron app mysteriously changes?
ie, regardless of anything the router and Lutron bridge might do, shouldn't a manually entered IP address in the Hubitat app be saved and remain unchanged until manually updated at some future point in time? Or is there some kind of attempted detection happening (I don't recall, haven't touched my own Lutron Integration app in a long time).
We will have to hear from him, because it was ambiguous what he meant. An app setting will not just change randomly, but the IP address assigned by DHCP in a router will. Inference: the latter is his problem.