Netflix streaming to TV from iphone LTE connection

Hello, I have been researching this for a while and cannot seem to find the answer (at least not ones I like).

I have an iPhone 5S with unlimited data (and hence no hotspot functionality). I want to stream Netflix to my TV.

Additionally, I have an Apple TV (with peer to peer functionality), a Chromecast, and a Sony Bravia TV with all the bells and whistles.

What is the best way for me to watch Netflix on the TV ? Keep in mind that any option that requires Wifi will not work because then I cannot use my unlimited LTE data.

To me, this rules out Chromecast, as it looks like it requires Wifi.

I haven't really been able to make the tv's connection work but I feel that it should (DIAL?). But, I suspect that using this method makes the Netflix app on the TV be the master and defeats the purpose as the TV Netflix app would stream the data.

Apple TV peer to peer seems like a no-brainer, BUT, the Netflix app will not reliably work with Airplay. I can make it work sometimes. Some episodes will reliably stream to AirPlay and others hang with a spinning wheel on the screen. It really seems like a Netflix issue, but whenever I talk to them and say I am using a LTE connection, they dismiss the problem as a bandwidth issue.

Any ideas would be greatly appreciated.

Thanks
Matt