Okay, I watched the video. Give me a second to recover from motion sickness. Your mouse is not a robot - would have been helpful to get an answer from you on that.
So, your best bet will be any stand alone ESP module equipped with a USB port. It will have to support HID, I have no idea how that is done on an ESP but I've seen it done with 32U4 based boards. Size doesn't seem to be an issue since it's not a robot or anything, in fact you could hide it in a box under the desk and it would work.
An example stand alone board is a WeMos D1.
A quick Google search tells me that the ESP does not have HID capability. So you need to pair it with some processor that can do it, like the Pro Micro or Leonardo, a few more.
Maybe you could manage to grab some idea of what hardware they used in the video. I can't, it would make me nauseous.
What is your level of hardware/software skill? I ask because a project like this is advanced.
This project automates video game play, by accessing the video game screen that is running on the PC, and transmitting coordinate information via wifi, to a wifi enabled mouse emulator.
There, that wasn't that hard, was it?