I have read some posts where a satellite box will reset very fast and will trigger from a transfer switch. Yes you can leave them on but it is possible that they would reset if switched to generator power.
BTW you probably only need 300 watts to run the tv/cable box. Much easier install. You can wire the transfer switch to run this circuit with inverter power as priority so that it does not switch unless the inverter is shut off. You could then run the generator and charge the battery or make some popcorn. For inverter I recommend sine wave only.