Simplest Solid State Tesla Switch

All the JBT transistors can be replaced with equiv. MOSFETs. I used voltage dependent current source and a resistor to mimick the behavior of a transformer. You can use 4 transformers instead in your implementation (MOSFETs are probably better than JBTs, I am affraid, as MOSFETs have no gate current). I used no diodes for switching purpose in this circuit. LTspice simulation is performed and the plot of the current over the load R3 is given below (I omitted the two capacitors at each end of the load R3 for the sake of simulation, in your implementation, you should include them).

This circuit should be the easiest to tune as well (Reason: when the pulse is at 6V, all transistors should be open switch, so logically, there will never be battery shorts). The pulse voltage source V5 can be replaced by a 555 astable circuit. V6 can be obtained by using an equal voltage divider (two 5K resisters, for example) sharing the same power as the 555 circuit. All the JBT transistors can be replaced with equiv. MOSFETs. I used voltage dependent current source and a resistor to mimick the behavior of a transformer.