Item 1: There are transients on the driver transistor bases
Cranking up the sensitivity on the 'scope probe placed on the PNP transistor's input pin I see a positive-going 100 mV transient as the adjacent grid goes positive. This is visible in decreasing levels as you view grids further away from the one going active, which makes sense. I'm guessing this is showing up on the transistor's base through the base-collector capacitance. At first I thought, "Ah ha! That's turning on the driver transistor!" Then it occurred to me that this is a PNP transistor, so it's actually reverse-biasing the B-E junction, keeping the thing turned off. The transistor is specified with a VEB of -10 V, so -100 mV isn't going to faze it.
Item 2: Zener diodes only work if you drive enough current through them.
I knew this, but I've never seen it quite so clearly. The behavior of the BZX79-C6V8 diode I'm using to offset the cathode above ground is specified at 1 and 5 mA. Even with one of the anodes driven to +30V wrt ground, not enough current is flowing through the Zener diode to develop a voltage across it. Driving the grid to +30V puts 6 mA through the diode and causes the cathode voltage to rise to +6.8V, but it drifts back down after the grid is pulled back to ground. This offset is needed to keep all the fluorescent segments dark when they're not being driven, so this needs to be fixed.
A simple solution to this would be to connect a 30K ohm resistor between the +30V supply and the positive side of the Zener diode (which is also the center tap of the transformer secondary winding). This will put 1 mA through the Zener at all times which will keep it happy and may also keep the boost regulator happier. Even with all the anodes and a grid driven I can't see the current through the Zener exceeding 11 mA, so an extra 1 mA won't approach its dissipation limits.
Unfortunately, I don't think this had much to do with the ghost segments, because I tried that and didn't see any improvement. I just didn't put a scope on it to measure the effect.
Item 3: I left off the capacitor across the Zener diode in the filament driver.
There's supposed to be a 10 nF capacitor across the Zener diode to conduct the switching transients around it. I didn't have this installed because of the way I laid out the components on my solderless breadboard. This also helps keep the voltage drop across the Zener constant during periods when none of the grids are driven high. But I don't think this had anything to do with the ghost segments as I wasn't switching digits or segments at the time.
Item 4: I found a filament transformer driver I like better.
The actual current measured through the filament driver IC and transformer is just under 200 mA at 3.3V. This means that I could use the Maxim MAX253 transformer driver with the Halo 4:3 transformer, though I'd be pushing it to its rated limit. But I won't, though, because I've decided I like the TI SN6505B better. Not only does it seem to be able to drive the transformer better, with output switch resistances 1/10th that of the MAX253, it has slew rate control and a built-in soft-start that may help minimize the inrush current when the filament is cold. It's more than twice the price, but I only need one.
While using the SN6505B and the Halo 4:3 transformer to drive the VFD filament, neither the chip nor the transformer get warm in any way. This means the heat I was seeing when using the 30 Ohm resistor as a load was, in fact, coming from the resistor and not the transformer. Unfortunately I can't find any spec for the maximum current or power through the Halo transformer beyond the vague statement that it's "designed for use with the MAX253", and the MAX253 is only rated for 200 mA average current. Alternate 4:3 center-tapped transformers are hard to come by in single unit quantities though. Wurth sells one rated for 1A but I can't find anyone who stocks it. Maybe they'll send me some samples.
The transformer does seem to ring just a little bit during the zero crossing points, as both ICs implement "break before make" to avoid grounding both ends of the transformer primary at the same time. I'm wondering if I shouldn't put a very small capacitor across the transformer secondary winding to snub this ringing as it shows up as noise on the the driver board. A 330 pF capacitor I had sitting on the bench does a nice job at the cost of 14 mW, while a 33 pF capacitor had no real effect. Perhaps there's a value between those two that will do the job for less power.