Admittedly I haven't tried the REST API or the Asset Discovery Tool (both of which I will when I get a chance), but I was pointed the right way by TAC and my account rep.
Network > Devices >
Use the drop-down toward the left to select "by Device Type" > Use the three-bar menu > Configuration/Firmware > Register/Export Serial Numbers...
This gives the full printout of all devices, blades, modules with serials. Previously, I'd tried that with the drop-down menu on "by Location" which did not give the full list.
