As long as you subscribe to any of their TV service tiers they're going to charge the Broadcast TV fee, even if you never use their service to watch those channels. The only way to avoid the fee is to cancel your TV subscription.
Your TiVo may be able to tune in cable or antenna, but it cannot do both at same time. You would have to reconfigure your TiVo each time to switch between the two of them. Who would do that? To make it clear your antenna channels do not filter into the cable channels or guide period.