You need a thing called "synchronous boost".
It's quite iffy to replace the diode only; you would need a very fast replacement circuit. Instead, normally you do this by picking a different DC/DC controller IC, one which is designed for synchronous boost. It will offer a gate drive signal for your synchronous FET directly!
TLDR: Replace IC1 with a synchronous boost controller which drives two FETs.
Another comment: Get rid of the discrete gate driver Q1/Q5, and just use a driver IC which can drive enough current to your MOSFET(s). At this power level, you shouldn't have problems finding complete integrated controller IC which has enough gate drive oomph so that no external gate driver (IC or discrete) is needed.