Is your cable modem/router a single unit? If you don't use a landline, the ends can be changed to ethernet. Then connect modem inside the low voltage box and put the router at the most convenient spot in the home. Then you can use the cable that currently runs from the low voltage box to the TV for your antenna cable.
Does Century link provide a modem and router at no charge? I assume you may have Xfinity/concast now. Do they offer a free router and modem?
I'm sure you could find someone that can help with the ethernet connections.
Splitting the installation of the internet hardware and using the current coax cable to the TV would be the simplest/cleanest installation. And if you wanted TV in all 3 locations, you'd just need a splitter and perhaps a preamp installed in the low voltage box. Splitting the cable at the bedroom to add to the back of the TV will decrease signal to the TV.