I bought an interesting wooden toolbox a few years back that is somewhat relevant to this discussion. Its similarities and differences may help you with your design.
It was originally built by/for a set designer for a traveling theatre company, so from the get-go it was designed to pack up and roll around as well as to fit conveniently in the moving truck. That ability to pack neatly in the truck heavily affected its design. It is actually two boxes, both about 24" square when closed up. The Bottom box is 36" high + wheels and the top is about 24" high, making it a cube.
Since I don't use it as intended, I have made some minor "adjustments" to make it better fit my needs. Firstly, I slipped a thin piece of plywood between the upper and lower boxes, and then screwed the top down to the bottom. That allowed the doors to open and close freely despite some wear and abuse that warped the top and bottom pieces before I got the box. I have found the 24"-deep hinged lid on top of the 60" of box and ~6" of wheel/castor height to be problematic when I used it in the basement, but not so much in the shop with the 9-foot ceilings or, presumably, in a theatre setting. I usually leave it propped up, but the top bin is pretty difficult to access so it is pretty much a waste of space.
Having two separate boxes has made moving this box much simpler and easier, and I highly recommend that you consider doing that as well.
As you can see, the 24"-deep base extends 6" more than the depth of the cabinet. This protects the doors when moving and aids in locking the box, but it also GREATLY increases its stability when rolling the box around, something that was no doubt an important design criterion in the theatre business. As for wheels, it had two large fixed wheels recessed into the back of the base cabinet and two swivel castors at the front. The rear wheels were obviously an issue, so they were modified like this before I got it. This new configuration makes it easy to maneuver while somewhat preventing random sideways motion on less-than-perfect floors.
In your case, since you will have a wider, shallower, and taller cabinet, pushing it around will make it less stable than this one. Changing the orientation of the fixed wheels 90 degrees to align them left-to-right would help make it more stable when moving but would also make it tricky or impossible to slide it in a tight space. The configuration that I have is ideal for that case.
All the drawers and shelves that my cabinet has are 18" (outside measurements from the back of the box to the front of the drawer). There are times when I feel that these are too deep. About 1/3 of the space behind the things on the shelves cannot be accessed without moving everything in front. The additional 6" of the bases has never interfered with access to the contents EXCEPT when I use it as a convenient place to set something down until I find a home for it.
Since you will not be having the doors (I hope! They are too shallow AND waste a lot of space compared to a cabinet), perhaps you could do three cabinets wide (12", 12" & 18") mounted between some top and bottom pieces similar to my cabinet with an additional cabinet stacked on top? This would be a similar configuration to modern mechanics' toolboxes.
I hope that this helps somehow. Please let us know what you and the customer decide upon.