At least when the leads are left relatively long, it is a good question if the LM399 is really that sensitive. Thermal EMF could cause some extra voltage, though this to a large part compensates due to symmetry. The other point here is whether the LM399 is that sensitive to soldering:
the LM399 runs rather hot and is know not show very little hysteresis. If the leads are long soldering might not even bring the pins at the case to normal operating temperature.
So there might be the option to use a rather normal socket for the burn in phase and than solder the chip directly to the board, with long enough leads and short solder time. For the LM399 I would not expect the classical reset of aging due to soldering.
If using a small daughter-board I would consider having the critical pins closer together (less thermal EMF) and possibly using separate current / voltage sense pins if plugs are used. So more like 6 pins in total. The connections could be soldered copper wire.