The simplest solution may be to use a buck-boost converter, and have it power the control circuitry at all times. Or if the control circuitry can safely run at, say, 11V, you might be able to just use a buck converter. Or buck the input down to 5V or 3.3V or whatever to power your main logic, and then boost back up to 12V for the bridge driver. Lots of options, depending on your overall power needs.
The question - how to properly do that to avoid high voltage passing by for a brief moment to prevent driver IC damage. Maybe P-MOSFET with a gate connected to DC converter output and a diode between DC converter output and MOSFET drain to prevent high voltage feeding back into DC converter output?
You would need to have the bypass transistor off by default -- just connecting the gate to the output of the DCDC converter won't work as you say. You'll need to keep both the DCDC and bypass off until the supply voltage stabilizes and you can reliably measure it to determine whether to use the DCDC or not. A voltage supervisor would be ideal for handling the turn-on delay, but a comparator with an RC delay could work. The problem with using the UVLO built into the DCDC is that you don't have access to the *output* of the internal comparator, so it's hard to guarantee that the bypass transistor is on when the DCDC is off.