Impressive that you got that low, though I'd question whether you were actually in regulation at that point. The datasheet says 'Minimum cathode current for regulation' is 1mA (450uA typ). Off state cathode current is 1uA (50nA typ), so I think you were probably right on the individual device specific bleeding edge of device turn-on.
That's because the cathode current was 1mA+
The load was not grounded. Here is the schematic I used (see attachment). Basically, I made a shunt current source. The load is in series with the shunt, and tlv431 connected in a way that if voltage drop on the shunt is more than 1.24V, it will "steal" the excessive current. The adj-pin bias current is 10uA max, so there is a potential to go even lower. I didn't check stability boundaries, but seemed stable to me under variety of voltages and currents and on different dmm ranges (i.e., different load resistance). At least my dmm didn't show any significant AC current (I used u61e).
I also checked ht7333-a connect as a current source (claimed to have 5uA typ quiscent current). Seemed to work well too, without any capacitors (a bit of grey area here, as, afaik, datasheet doesn't discuss stability). Checked down to 65uA using 50k trimpot.
My next candidate is LM317L which has quiscent current of 100uA max. This IC, afaik, quite stable, even without external capacitors thanks to darlington emitter folower architecture. I'll check and report it here. I used this little device in the past and I was impressed with its performance. It seemd to me much better than lm317, at least reference voltage so much more stable. I evaluated devices from TI, on semi, and, afaik, ST, all showed good performance. I'll also evaluate min dropout voltage, as it seems the major drowback with lm317* devices. That's because of darlington output stage (with, afaik, three transistors) plus drop on current sense resistor.