Is there any good way to limit primary current on a low inductance transformer?
That type of flash-cap charging transformer will invariably be a flyback.
You can reduce the power throughput, and hence the average current from the power supply, by lowering the pulse repetition rate. With a low primary inductance, the pulses will still need to be short. However, you only need to repeat them as often as is needed to get your power throughput. It's not a question of simply reducing the frequency to 200 kHz for instance, you need to define or even control the pulse length and repetition rate.
With a decent input capacitor, you might be drawing a high peak current from it during the charging pulse. However if the charging pulses are few and far between, the average current into the capacitor from your supply would be quite low.