The charge line from your TV will supply 12 volt
power to your coach when the TV is running or you have the key on.
Unfortunately the wiring is too small to work well. Plus, you will drain the TV battery to the point where it may not start the TV.
The converter and TV are wired in parallel in order for either to charge the coach battery.
You will need a battery in the coach to provide power to the breakaway switch when towing. In case the coach becomes unhitched from the TV. Most, if not all states require this.
Deep Cycle batteries are the best choice for your coach. The battery in your TV is designed differently for high current short period starts. A typical "start" battery is not the best choice when it comes to providing 12 volt
power to your coach.