You might be able to do something like this:
Create two rigs, one for the skeleton, and one for the constraints (you can create the constraint rig out of one object by using the Custom mask feature in physics). The model itself will have no physics behavior at all, it's parts will simply be set to the x, y, and angle of each of the appropriate skeleton pieces. The rigs and the colliders will all have physics on them.
Hinge some colliders at the extremities so the movement of the skeleton is limited by the constraint rig. Physics behavior in Construct has an option to disable collisions between certain objects with an action, so you will disable collisions between the skeleton rig and the constraint rig so that they're not fighting each other. Only the colliders will be affected by the constraint rig, and they will keep the skeleton from moving too far.
I haven't tried anything like this myself, but I'm quite sure that with some tweaking it would work.
As for the mouse movement, I'm sure I have an example of how to blend mouse and physics around here somewhere... I'll have a look.