Hi!
I'm running a nearly identical setup and I would say you're on the right track but the main difference is that my whole setup is 5V which makes powering everything simpler. Is there a particular reason for the 12V LEDs? For a 55" tv, a 5V SK6812RGBW strip powered at both ends will be fine - my setup is around a 125" projector screen with four power injections and it works great, no issues with voltage drop.
This will allow you to use one 5V power supply for everything (calculate your needed amperage of course) so you can more easily ensure common ground between all devices - with a single 5V power supply, I would also add a ground between the PICO and the rPi. I'm not an electrical expert so I don't know if a common ground between 5V and 12V power supplies is safe, or a good idea.
Also, SK6812RGBW can be very troublesome to get working properly so anything you can do to simplify the setup is probably a good idea.
If you have any other questions I'll try answer best I can.
Adding a couple photos of my wiring in case it helps: