This is probably just HDCP (do a Forum search, there are hundreds of posts about it). Just remove the HDMI cable COMPLETELY and feed the TV from the component OUT plugs on the Slingbox.
And you are wrong about the effect on the TV screen. Your DirecTV box will output ONLY one channel on all outputs. If you want to watch a different channel on the Slingbox you will have to have a second satellite box.