Hi, that's looking pretty cool!
First, the TVs have a soft off mode that lasts for about 2 minutes. In that time, you can use the remotes to turn them back on, but after that you need to press the top button/switch.
Next, there's no way to bypass that switch other than by physically soldering wires to certain pins on the switch and connecting your own. If you are not familiar with soldering then I do not recommend you try this as you could permanently damage the board.
An alternative to all of this would be to modify the firmware to never fully shutoff, just turn the screen off, like in soft off mode. This way the remote could be used to toggle them on/off. Obviously, the TVs will always need to be plugged into power since the batteries will die.
If you are familiar with programming and Arduino, you can find the code here:
https://github.com/TinyCircuits/TinyCircuits-TinyTVs-Firmware. It looks like you would have to remove these lines:
https://github.com/TinyCircuits/TinyCircuits-TinyTVs-Firmware/blob/0940ebeb370e1ede4d30c0969dea52f89c328774/TinyCircuits-TinyTVs-Firmware.ino#L261-L263