An (active) HDMI splitter will work, but a Y-cable is unlikely to esp given you want to run it all the way to the bedroom. Being a digital signal, HDMI will work or not at all, you will not suffer picture quality loss. That said, if it is on the edge you will see picture breakup, but that will be quite obvious.
How do you plan to get the IR remote signal from your bedroom to the TiVo? You could get an 'IR extender'.
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