If you mean to integrate the power switches, and both operate between
the primary negative and positive rails, then you don't really have a
driver other than the taper-chain and anti-shoot-through. 5V POL DC-DCs
are that way (although in the interest of compact die size, NMOS high side
is often a "better deal" aside from requiring the customer to add a bootstrap
capacitor to their BOM).
I find your description of various voltages and actions unclear / messy.
I think some time spent block by block, from the outside in, would help
you make sense of things (rather than collecting random nuggets of
Internet advice).
As an example - you know the load and the edge dV/dt you want, to
support the fSW and overlap interests, right? So now you can size the
final NMOS and PMOS. And from there, size the last predriver stage to
not-degrade (4 devices, with ideal gate drive phasing to make-so the
slew and nonoverlap. Then the eightt devices (4 pairs) preceding...
It's all just a big dumb inverter, real big. Until you start adding stuff
like anti-shoot-through (servo or ballistic), output behavior in shutdown
and current sensing, anyway. Which all had best be on the table from
the start, as otherwise you'll be taking another lap around that obstacle
course.