How to research flicker free controllable continuous RGB light sources?

I'd forget domestic lighting altogether. I doubt anyone is going to bother making it flicker-free because it's not on the list of requirements for your average household.

Google, eBay, Amazon etc & search "video lighting panel RGB" for many options, from $£€ 50 upwards.

The flicker or banding is caused by the light pulsing while the sensor rows are scanning across the frame.

It is possible to reduce this if you have your speed at multiples of the electricity Hz. In the US, AC electricity is 60 Hz, so that is 1/60th a second. This may not always solve the issue though.

For photography, if your camera allows it, you can use the mechanical shutter curtain instead of any electronic shutter curtains. Not all camera models allow this. That can stop the amount of light the sensor receives, as the data continues to scan out to storage.

Some LED lights (like professional studio lights) may use DC instead of AC. They may also use voltage adjustment instead of PMW. The youtube channel DIY perks discussed a little about how to construct such an LED light panel that doesn't flicker. That may provide some ideas of what to evaluate when looking for a light.

The only other thing to do is to seek incandescent lighting where the filament heats up to produce light. This does not flicker even though it is on AC. Incandescent lights in the US are on the way out, but you may still be able to find some 60w or 100w bulbs. The other thing you can use is a halogen bulb inside a halogen fixture. Many professional tungsten lights of yesteryear use halogen bulbs. You can find them on ebay these days for much cheaper than they used to be in the past. They use a lot of power and the bulbs are fragile, and can be damaged if not handled carefully. They will always burn out, but these should be around for some years to come, and you won't have flickering or banding with them.

I would look for video lights and start from that premise.

If they are dimmable, they will include a refresh rate good enough for 1/60 at least (for video) and a decent CRI.

One problem that I see is the limitations to be RGB because I doubt they will give you a good CRI.