I am starting to understand why I have luck with other similar power supplies, but this type has been problematic.
A little more background on this unit, The transformer, capacitors, resistors needed to accomplish what is needed are all potted and not serviceable. The unit is meant to be portable and operates off of a 6 volt battery. So a bipolar bench supply is out of the equation.
Attached is a schematic that has been cleaned up removing the voltage divider, which should not effect this discussion.
The blue box is the potting and you can see there is no way to separate the ground of the primary wingding from the ground. The two capacitors in series I believe are just to filter.
The purple box is an external mechanical vibrator that works a little different than others I have played with on old car radios, portable two way radios. These units have center tap transformers.
There are electronic replacements out there, and I have bough or build many of them, and they work in other power supplies, but not this unit.
Trying various other mechanical vibrators I have found they get hot quickly. These are all 80 plus years old, inefficient, and many just no longer work. As mentioned the correct vibrators can be too slow or two fast and create voltage spikes.
this vibrator when energized takes the path of least resistance to the internal coil which pulls the read from the lower connection to the upper and puts power to the primary side of the coil. As this happens the current is limited by the internal resistor which releases the read where it begins the cycle all over again.
As I understand it the vibrator is doing nothing more than pulsing the DC to the primary.
This is where I got the idea of a 555 timer.
There may be high voltage feedback involved.
I am unclear what the .1uf capacitor is doing. is boosting the primary or vibrator coil? And to what end? or is it as Andy said "you could try a large 1000uF non-polarised capacitor* in series with your transformer winding. This circuit topology is simple, but generally low power." and this helps with the saturation.
Or maybe saturation was part of the design. I say this as the HV output is only a few micro-amps but it can zing you at the battery.
Looking into the H-bridge, would this even be possible as it would be reversing through the ground which might upset the voltage dividers (not shown)