I don't have a HV probe (been meaning to buy or build one actually), but I think I should be ok testing the signal into the Z-amp and the output out of the Z-amp directly with a DMM as it should only be a maximum of 120Vdc (I'm assuming this is floating at the 1550V? But I could take a relative measurement?).
...is this correct? 
Based on experience with other scopes (i.e. there might be significant differences!)...
The "Z-amp" in that block diagram will be in two sections: low voltage (i.e. a few hundred volts from ground) and high voltage (i.e. a few kV from ground). A "blanking waveform" will be created in the low-voltage section, and then its AC component "copied" up to the high voltage section by a "DC restorer" circuit. The grid-cathode waveform should be the same as the blanking waveform, offset from zero by the cathode voltage.
You can debug the low "voltage blanking" waveform using standard tools and techniques that can withstand a few hundred volts. You cannot do that directly at high voltage. What you can do, with care, is use a handheld DMM to measure the AC voltages in the low voltage section for various combinations of brightness settings, and then see the same voltages are visible in the high voltage section. If not, then the DC restorer is faulty.
Personally I would suspect that your DC restorer is working fine, but that the blanking voltages created in the low voltage section are faulty. There maybe potentiometers in the low voltage section that allow the voltages to be tweaked.