Why doesn't commercial LCR meters' band start from very low frequencies?
They are custom made. For example, this one which works down to 10&nbsp;µHz (100,000 seconds period, 1.2 days). 

But they're not commonly needed, not in electronics. Sometimes in material science measurements. When we had to make measurements of permeability of a special ultra-pure metal we just threw together an experimental rig with lock-in amplifier, power driver and some other instruments we had laying around rather than spending tens of thousands of dollars on a specialized instrument, probably similar to what you are doing. The very low frequency was necessary because of eddy currents. We made vector measurements and extrapolated down to extreme low frequencies.