I've decided there is plenty of space for integrating bits storage to the body itself. It uses short M4 inserts as the holes are not through, making them easier to print. Longer inserts shout fit too, there is a STEP file if you would like to make these through holes.
Other than that, that's the exact same file as the original Body part.
The author remixed this model.
Added bits storage to the body of the press.