The task is common, but not to the tolerances you seem to require. The concerns are more typically preserving dihedral states and chirality with restraints during equilibration, and relying on the balls and springs force field for the rest.
One option might be defining custom atoms for your target molecule, so that you can increase the force constants for the bond, angle, and dihedral terms to achieve the degree of stiffness you want. However, this would require a non-trivial effort.
As far as I know, only the SHAPES facility was intended for rigid body dynamics, but I don't know that much about it.
If you wish to allow whole molecule translation and rotation, CONS HARM BESTFIT might be an option instead of CONS HARM ABSO, but I doubt it would achieve the tolerances you want.