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I suppose that it doesn't have to be HD, but it does have to be more powerful than what you have. But today, all cameras that are currently being manufactured are either HD or 4K capable. But for the best result, you need to make sure that the chroma sub-sampling ratio is 4:2:2 or better. 4:4:4 is the best, but you will fork out a pretty penny for that.
Your green screen set sounds like it is adequate for what you want to do. Although, it may not be necessary. You can shoot your scenes multiple times, placing yourself each time where one of the band members will be. You can do this without the green screen, and use the background that you want. Just be sure that you never move the camera. Then, once you have each clip in the timeline of your NLE of choice, you use the entire frame of one of the clips, keeping it on the bottom track of the stack of clips, and then put the other clips in their own individual tracks. You then mask around each other band member in the other tracks, so that the footage from the tracks below shows through the masks. This way you won't have to worry about how well your green screen is lit, or having to pull a perfect key, as it takes the need for a green screen out of the equation. It would also mean that you won't have to worry as much about the chroma sub-sampling ratio of your camera. This is actually a fairly easy effect to pull off. Let me know if you have any other questions.