I'm not sure I understand the problem described in the report, but I can confirm that the duplicate spline does not become active.

The original code complicates a lot and unnecessarily.
For example, when executing calc_duplicate_actvert, the value of cu->actnu is temporarily an index greater than the number of splines.
This index only becomes valid after calling BLI_addtail(newnurb, newnu) and BLI_movelisttolist(...).

Also the purpose of calc_duplicate_actvert is just to check if the index of the active vert is within a range calculated outside the function.

And, in the other function separate_exec, this calculated value for cu->actnu is completely ignored and replaced.

The code would be much simpler if the active vertex and active Nurb were referenced and BKE_curve_nurb_vert_active_set was called right after BLI_movelisttolist.

But that would be more code cleanup than fixing, so for a quick fix I believe this patch is valid.