We start with our sine voltage:

Full wave rectify it:

And feed it to the cap. Without a spark gap, the cap voltage (shown in green) would follow the input and charge to max voltage and then stay there (ignoring leakage) like so:

If the spark gap were set to fire at the 0 Volt point of the transformer and the spark were theoretically 0 ohms (does anyone have any idea of an approximation of the resistance, or conductance of the spark?) the voltage on the cap would appear as in this illustration. Notice that the duty cycle on the transformer just dropped to 50%.

Also realize that this would not increase the current that the transformer uses to charge the cap alone! If we assume that the spark only transfers 1/2 to 2/3 of the voltage to the tuned circuit the wave would look similar to the following illustration drawing even less current (duty cycle <50%) from the transformer.

This is what I am thinking of doing. C1 would charge on each half swing of the output of  D1 and L2/L3. L2 and L3 will be calculated to have minimum reactance at 120Hz to allow charging and a high reactance at about 200 KHz to stop feedback from the tuned circuit. The synchronous spark gap (SSG) has to be timed to fire at the 0 Voltage points of the L1 output. If not, I can see going through lots of rectifiers (D1)(rectifiers are better fuses than fuses)! For safety sake, I wonder if the TC primary circuit should be left floating, or if the secondary of  L1 is center tapped, should it be grounded? It seems to me that keeping it floating is the safest bet - The pole transformer should be amply isolated, primary to secondary so that even if an arc-over should occur, the 13,800 cannot complete a path to ground.