Can I use 4G LTE (Cellular Data) and Airplay using Wifi w/ No Internet connection at the same time? I've looked for an answer to this but still haven't found one that works.

I want to use my iPhone as my primary source of internet. I am currently an ATT Uverse DSL customer, but I want to cut the cord and use our current DSL modem/router, the one and only Motorola NVG510 for it's wireless signal only so that my wife and I can airplay from our phones to our Apple TV. 

Is it possible to use our Cell Data to stream a video and then airplay that streaming video to our Apple TV at the same time?

*Using the iPhone as a hotspot is not an option.
See my response to this question on the Apple discussions.

https://discussions.apple.com/thread/5770041?tstart=0

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.

Edit:  I also notice you said use it as your primary Internet connection.  I originally assumed that you meant only for the iPhone.  If so, then my response above should work for you.  If you meant use it as your primary Internet connection for all other devices (laptops, etc) in your home, then of course the only way to do that IS to use your iPhone as a hot spot.  That's kind of the definition of a hotspot.