((I used a Linkit 7697 and this simple circuit:))
It is a bit more complex than I was thinking about.
Well maybe not as complicated as you might think. I could not see your remote very well in the video, but the ones that I have seen look something like this one:
https://www.amazon.com/SUPERNIGHT-Controller-Compatible-Assistant-Working/dp/B08HVVRPBH/ref=sr_1_20?dchild=1&keywords=IF+Rgb+Controller&qid=1603921702&sr=8-20Basically, there are three LEDs, R, G, B and using PWM you vary the duty cycle of each - that can get you to a mess of colors. So, you need to find the codes for selecting R/G/B and then the ones that bump the duty cycle up or down (likely brightness +/-).
So, if one color is
B R=70%, Green=20% and B=50% and you change the PWM duty cycle on blue - by sending the command to increase brightness (duty cycle). It is probably the case that each brightness command increases/decreases duty cycle by a set amount - say 10%, so you would need to track that and provide multiple commands.
Going to preset colors is the same thing, but easier. You could go to duty cycle=0 for all three and then step them up to their presets.
The whole issue of color mixing with LEDs is fascinating. Some years ago (probably 10) I heard about PWM and a little PIC chip (12F1572) that had three 16 bit, PWM registers and I spent a ton of time learning with that chip. I made a "party light" ooooooahhhhh using just that chip and a couple of batteries and a step up. It goes through Roy G. Biv and some other "scripts". Works great, BUT mixing colors smoothly - going from orange to indigo for example is not trivial and there can be more math than my aged neurons can entertain...still it works and I think you should play around with your set up - you have the vision already, now you just need the code and parts to cooperate.
Of course I could be all wrong about this