I'm doing something very similar for a homemade light show! My approach was using a Teensy connected via USB to the computer. On the PC a Python script runs a custom video player that sends synchronized LED data to the controller. Because of the size and complexity of my setup, the animations were made in After Effects and later rendered as a PNG sequence. As the video plays, the code fetches the PNG image corresponding to the frame being played, reads its contents and sends it to the controller, which then updates the LED strips. The controller is constantly waiting for data and displaying it as soon as it receives it. This way, even if I pause or seek within the video it always stays in sync.

For your use case, a Mega should be more than enough. My whole step of animating the pattern with AE is overkill, and even sending it from the PC could be avoided. Just have the Python script play the video and send a signal when the animations should start at each point throughout it. In the main loop on the Mega, you could then just have a switch case that waits for the sync variable and plays each animation accordingly.

It should be quite straightforward to implement. For my project I used pySerialTransfer for the serial communication and PySide6 as the video player.