You misread again
The driver is referenced to the transistor source with a floating supply; it should be fine.
It's not clear if the 200W dissipation is due to including startup transients in the calculation. 10ms is a very short duration to average over, in realistic terms! You simply need to select the steady part of the waveform and do the power calculation out there. (You didn't show the power calculation so I have no idea if you made this mistake, or did all this already and still got the large result. Just guessing here.)
I have a different complaint, however:
This is a naive circuit -- while PWM is part of the operation of a switching converter, it's not what we're doing; it's secondary to the intent. What are we
actually doing? Switching inductor current. Why? Because inductor current dictates transistor and diode dissipation, and if that current is allowed to hit crazy levels (like during the startup transient pictured!), transistors and diodes will explode!
So we prefer circuits which control PWM, to set inductor current, and then we can just about guarantee our transistors/diodes will survive even when the end user accidentally shorts the output. We don't care what the PWM% actually is, as long as it gets the inductor current where we want it.
This is called current mode control. (The fixed-PWM example is called "open loop", while a controlled-PWM circuit is called "voltage mode". Both are inferior, or require protection circuitry hacked on, just to emulate what a current-mode controller does from the start.
)
This is kind of hard to do with a 555, but it can be done. Here's an example in the boost topology:
https://www.seventransistorlabs.com/Images/555%20Boost.pdfNote that R13 is a shunt resistor that senses Q1's drain-source current, and Q3 (relative to the threshold set by R4+R8) measures that current, and shuts off IC1 when that current goes above the setpoint. The setpoint is in turn determined by IC2A, the error amplifier which ultimately regulates output voltage. This is all a bit convoluted I'm afraid, hence Q2 and support components -- it's a forced fit to make the 555 operate this way, and there are much better ICs out there to do this same job, like UC3842.
For your application (inverting buck-boost), most controllers will need a high side gate driver and supply. This isn't hard to solve, but does increase the component count. Alternately, a more common buck controller can be used, in "bootstrap" connection, to make the negative supply. Or a Ćuk converter (which uses a double inductor cleverly) can use the same boost circuit, while creating a negative output.
Tim