We use a Comcast wireless box in the master bedroom and it works as well as any of our other X1 wired boxes. So assuming you have acceptable WiFi coverage on the lanai, a wireless box would be the way to go.

You should not need a cable connection. If your wifi goes to your lani you should be able to stream your live tv using the xfinity app on a roku stick or another streaming device supported by comcast

Comcast Statement on Xfinity TV and Streaming Devices

I just received my FREE Comcast Streaming Box. Either it or the service is called Peacock. The first box is free and more boxes lead you back to equipment rental of only $5 per month per box. It works with my Motorola Modem/Router. It will not have all the features functional until 7/15/2020 but it does work on a limited basis now and maybe the answer to your situation. Good Luck. It appears to be a winner.