I believe that the high-side MOSFET should connect to the positive power rail rather than ground, and the other terminal should connect to 5Vout via an inductor. That's the only configuration that makes sense to me.
I also believe that OUTA/OUTB of UCC2808A should connect to INA/INB of the mystery IC.
What is the supply voltage for UCC2808A? Is it the same as the mystery IC? I expect it should be around 10V.
BTW, I searched for dual gate drivers in 6-pin packages and found only a few. None had the desired pinout.