OVKTD: My rebbi once said the ‘mekor’ for bouncing after the three steps forward is very simple. Really, you’re supposed stay where you stepped back to until kedusha when you step forward and bounce at the appropriate points. Somehow it got into people’s heads that it has something to do with stepping forward.
Sam2: Whether the sefer chasidim says it or not, I’m pretty sure the mekor is the RaMBaM. The shulchan aruch generally doesn’t bring the sefer Chasidim.
The reason it is important to stay bent the whole time is because “oseh shalom” is the fifth bow of shmoneh esrei (which the rambam says should be at the end of sim shalom, but we push it till after the “techinos” of elokai n’tzor) and if you get up in the middle you are bowing too many times, which the gemara calls yuhara.