reference voltage (+15.000V) seems to be ok. The variation on the raw reference might be due to the DMM, as it's a sensitive high impedance point .
34401A should be in the Hi Z mode, and in 6 digit, slow mode, when measuring precise voltages.
I would not test at all ACV ripple at 1kV dc!! Might damage your 34401A!
Maybe you test up to 500V only, because HV is switched from different transformer taps, so limiting the danger .
If you have set 1000.000V, and you measure 982V only, then there's a different bug than some kind of ripple.. the regulation fails.
What about the -15V auxiliary supply voltage?
If 10V range is already unstable, you should concentrate on that range at first.
What's the output, when you dial 10.00000V?
You should check the chopper amplifier, if all voltages are present as expected, then the series regulator, and the protection circuits around.. the expected voltages should also be clear from the schematics, I hope.
Also check, if the current limit indicator (light bulb ok?) is on or off.. maybe this circuit is always limiting, which may lead to unstable output voltage.
Frank