The "showing the camera your tech cards" worked out pretty well, actually. Some minor issues with the sleeves reflecting light from the window that make it hard to make out some cards. So long as we angle the cards slightly, so they're not shown full-on to the camera, it should be fine. The camera's vantage point is actually good enough that you should be totally able to capture each player's hands, if they just tilt a little at the camera at the start of their turn!
I do need to see if I can get any additional height next time, because as it is both player boards just baaaaarely fit into frame. If I can also move the tripod and camera off the table, that'll get rid of the occasional blurring as the table gets bumped. But it's a good start!