@SONB I believe the issue is that you are actually calculating acceleration, not velocity. At step 1, the velocity vector length is already in units of uu/sec. Your step 2 puts it in units of uu/sec^2. So all you need to do are steps 1 and 3 to get m/s.
When I print the velocity vector length using the Start Circularized option to create as close to a circular orbit as floating points will allow, the velocity stays very steady for me in each physics mode. When not in a circular orbit, the velocity does change as expected.