Our teams are actually working to find what is the cause of this behavior and has been fixed for some setups in the most recent app updates. As you mentioned, if you are on the same local network, the 10-minute timeout should not happen.
However, it is seeming difficult for our team to replicate this behavior to see what is causing the disconnect. We have a few clues as to what may be causing the cameras to either incorrectly give the prompt, or switch to mobile data and not switch back. If you can, can you provide some additional information regarding your network setup and wifi?
If possible/known, can you provide:
Type of Router + Model
Is this a mesh network, or does it have separate SSIDs?
Do you have any APs or Range Extenders connected to the network?
What Type of Phone are you Using?
What is the Version of your Current Kasa App?
What band of your network is your phone connecting to, 2.4Ghz or 5Ghz?
Do you have a VPN connection or Private Addressing enabled on your phone? This could cause the prompt when your device is roaming betwen bands/nodes of your network.