Basically, use a static configuration for your WiFi Network, but leave the Router IP address blank. The LTE icon will be displayed, and the mobile network used for all non-local traffic (ie, Internet) but you will still be connected to the WiFi network can can use all local LAN resources.
Get a Generic router or access point, connect your phone to the wifi signal (sometimes it takes a while because it says no internet, after a bit it asks you if you want to continue without internet, say yes); connect your apple TV to the same access point. On your PHONE, go into Settings - WiFi, and select the network you are connecting to without internet, then select the little 'i' on the right. Then select "Static". Set IP address to 10.0.1.4 and subnet mask to 255.255.0.0
Leave all other blank.
This will ensure your phone looks to cellular for internet connection, but allows the wifi necessary for AirPlay. Just make sure you have a fast enough wifi access point/router or you'll have trouble. Also, some apps WANT wifi internet to work right.





