I'd suggest just painting the bottom of the pot, not the sides at all.
There's a concept of "view angle": How much of your view sees hot versus cold objects.
For the pot looking down, 180 x 360 degrees, there's some pretty hot stuff - hot steel bits, campfire coals, etc. The objects around the bottom of the bottom are warmer than the pot so it should be black so it absorbs more radiant heat.
For the side of the pot, they look out at the same terrain and scenery and sky that we do. Most of the objects being ambient temperature, the clouds cooler, depending on their elevation and deep space on a clear day or night, quite cold (4 degrees Kelvin, I think). The sides' surroundings are colder than the pot, so it should be left shiny so it emits less heat.
A pot within a windscreen probably should be painted black up to the height of the windscreen. Let me puzzle out how to answer that particular question.
I suspect that painting the bottle black will help the most on a white gas stoves, less on a butane stove and much less on a alcohol stove. But I'll be testing that in a few days. I just bought four 4 identical stainless steel canisters as Walmart that are comparable in thickness and diameter to BP pots and I'll modify three of them in different ways and try them on different stoves.
So for now, if you're leaving for a thru hike tomorrow, I'd say: paint the bottom black. If you've leaving next week, I hope to have more definitive answers.