LSB value at 5V reference would be ~150µV. All your external influences on DUT should be as low as possible, but at least you should decrease it to less then half of this value.
This would include noise, reference accuracy and all other possible influences. This is not trivial task at all. HP PSU is probably better then any regulator you could come up with, but you would still need to decouple DUT power and reference and reduce BW of reference as much as possible. HP PSU gives out 2mV p-p noise in 20Hz - 20MHz. Cabling to device, shielding it, isolating it mechanically. avoiding thermocouple effects, voltage drops on lines,...everything has to be done properly.