Inverting Charge Pump with Push/Pull Stage
There's continuous base current route for Q3. It goes either via Q2 or R2+Q1 and it doesn't stop at all. Q3 conducts at the same time as Q1.
ADD: no more that fault, the questioner fixed it in the question.
There's another error. The storage time of common emitter switching transistors makes Q3 and Q1 still conduct a moment at the same time. Storage effect means the Ic continues some time after the base current has been stopped. It can cause no special harm when the states change sparsely, but it causes easily high dissipation when the frequency is 10kHz or more.
I guess you are not going to accept the usual emitter followers due the voltage drop. Emitter follower would be free of the storage phenomena and a common driver would work for them. In common emitter switches you must limit the base current overdrive for ex. like in Schottky TTL, have speeding capacitors, have base discharge resistors and (as you seemingly already have noticed) have separated drivers for the output transistors to prevent the unwanted continuous base current path.
Bootstrapped emitter followers can do the job without too much voltage drop but that's essentially like adding a voltage booster to get more than 12V. That would allow the 1,5Vpp drop caused by the emitter followers and still leave nearly 12Vpp output.
Your own attempt can be developed a little. Here's an example:

The transistors are 2N3904 and 2N3906. V1=0 to 5V squarewave 33kHz, V2=12VDC, the output can pull 80 Ohm resistors to rails within 0,35V, the output swings from 0,35 to 11,65 volts.
Some attention is paid to prevent excessive saturation and you see there's also a speeding capacitor. No Schottky clamp type tricks are used. For higher frequencies you maybe should use it and find higher spec transistors.