Do the temperature coefficients of the resistors in the LM399/LM199 "Portable Calibrator" significantly effect the output stability?
Well if the LM399 has a 0.5 ppm/'C spec 0~75'C that's not good enough for 6.5 digit DMM with 1 ppm accuracy.  However the LTZ1000 is 10x better at 0.05 ppm/'C.

It is already thermally heated inside with thermal feedback.  Normally better stability can obtained with a double thermal oven servo over the chip just as done in better OCXO's.

Getting laser trimmed Resistor ratios to 1 ppm is a harder task. 

Never assume the accuracy is the same as the resolution. Sometimes the extra resolution in the short term is what is needed.

Here is a 7.5 digit meter with 50 ppm accuracy  ( only !)


Here's a test that compared references

Keysight 34498A ... innards of a 6.5 digit DMM


Keysight's Truevolt 34465A DMM, the 1-year specification applies for temperatures ± 2 °C of the calibration temperature and with self-calibration achieves an accuracy in the 10V range within 24h if 10 ppm of reading and  4ppm of range.

The only other Keysight DMM with ACAL is the high-end 3458A with its high-end price;
a bestselling 8.5-digit DMM