To stream we would have to digitize the HD content, then get it to some streamer and then another box to unstream it back to HDMI.

Sounds expensive.

Why not a HDMI splitter (no, these are not cheap either but cheaper than above) and a long cable?

And we can try wireless HDMI if we want to cut the cord down a bit. Example at

There may be other solutions like this but between cables and boxes, figure under 400 bucks.