Your carpet is more then 3/8 thick. you will basically be the same height wise as the carpet. I would do the flooring on the steps, to match with the flooring you already have, but that is my preferrence. Some like solid treads

You will actually want at least 2 tools for this job if you dont want to take the current tread off the stingers. a jig saw or circular saw will work to get 90% of the cut, but you will still have the last 3 inches to cut on the skirt side (unless the stairway is open on bothsides) for which you will need something more like a Fein Saw, or an oscillating saw. a jig saw or circular saw alone wont get tight enough. and you could use a fein saw to do it all, but you will use a TON of blades and time.

I think the proper way is to remove the balusters, cover the entire tread and reinstall the balusters. I can't imagine a solution where Bamboo in the middle of a tread looks good butted up against a false tread end.

Is there real hardwood anywhere else in the home? If not I wouldn't introduce real wood around faux wood. The two will never look good next to each other. Stick to the carpet, so save up to replace all of the flooring