Here is some data on an LTZ1000 ref I've been playing with, this does not look very good does it?
Frankly speaking, your whole set-up, including the cables, is not looking good at all.
I also do not know, what's inside the aluminum sheet, or what your LTZ reference you're actually using.
Anyhow, you're always comparing the LTZ versus your DMM, so the noise is always a combination of both.
Your instrument should perform noise-wise nearly as good as a 3458A, the whole setup seems to have too few resolution, at least the rms noise for an LTZ is on the order of 200nV
rms or 0.03ppm only.. maybe you also use Standard Deviation for measuring the noise.That 1.7µV
pp is not optimal, but on the order of what to expect.
If you want to have stable measurements, you're really dealing with sub-ppm stability - what a properly built LTZ can deliver, you have first to set up everything in a stable way, not that Flying Dutchman approach.. these loosely hanging, un-shielded cables with inappropriate connection jacks are simply acting like antennas, and by themselves create a lot of other disturbances.
If your actual LTZ circuit (?) is built from the original LT schematic, then it's not inert at all against E.M.C., so the hint for SMPSUs from others, is the crucial point here.
Maybe you understand now, that 'better' components, layout, and so on are not the key points.. maybe you just do the basics first.
I will append a picture how the stability diagram will look like.
Frank