I reduced both the base windings (4 or 5  turns less, 4 turns remaining) I could increase the collectors winding turns, but I havent.

Now I have safe voltages on the base.
Still the transistors become hot. I can balance the base voltage so both transistors become equally hot.
But once the psu voltage changes, I need to rebalance with the base resistors again so the load is shared.

it is time for a heat sink.
I prefer having both transistors on the same sink, so they have the same temperature, but that is not practical now.

scope shows base voltages under 5V

I placed small heatsinks with pasta on the transistors.
still they become hot quickly with small amounts of power.
base voltages are now good, under 5V with the reduced windings. a simple effective solution.

If I want to reduce the heat, at higher power levels, I would be wise to switch to mosfets, with low Rds

I probed the signal at the commons, before the diode, it shows around 10V with load connected, and around 0,7A 7V power

Still I see no benefits for the battery.
The battery impedance is no match for the impulses, and dampens them instantly.
And a lot more power is needed, to get a decent result in charging (or it will take long time to charge)
 
I think I'm done with this. it was fun. The dual inpulse idea works. but... I dont have any use for it.
battery can be charged, but ... heat means energy loss. better charge it directly with DC