Home Theater HDMI Splitting
An HDMI splitter will work, though from the ones I've looked at, they have to send the exact same signal to both. So if one device is at 720p and the other 1080p, both will be forced down to 720p (HDMI is a two-way protocol, so I assume the splitter has a chip to negotiate this sort of thing). If only one output is on, it should work normally.

A simpler option for you, if you aren't planning on using both screens simultaneously, might be an HDMI switch, which just switches which output receives the input. Make sure to get one that's bi-directional, as the common use case for these would be switching the input, not the output. There are switches that have IR remotes which would work with your Harmony Hub.