My friend and I have been building just such a program for a while now. We are both self employed software devs and we didn't have a clean way of tracking time, so we started building something that just did that at first, but it's evolved a lot since then. However, it's still very much in development so customisation is limited, but the core stuff all works. Now granted, it doesn't use docker, it's just a stand alone program and currently doesn't have cloud support (we are working on it), but I can absolutely send you a copy if you wanna try it out.

Exactly what I was thinking. Looks awesome!