I think the only reason for the LM3999 was for NS to be able to offer a heated zener reference in a cheap plastic package. The shared ground current between the heater and the zener is a problem due to lead resistance, solder joint, and PCB routing. It's also non hermetic.
As Kleinstein says, fitting an LM399 and running a separate heater ground wire back to a non current sensitive 0V point would be easy. The only requirement is that the heater ground must not go above the zener ground by more than a fraction of a diode drop (the parasitic substrate diode on the LM399).
As long as the two zener pads are maintained at equal temperature, there shouldn't be any additional thermocouple effects from adding a thin heater ground wire.