A Schottky is not usually necessary or worthwhile.
If D1 is implemented in an integrated circuit, it is likely quite fast and reverse recovery current & charge -- while they can be large -- won't exceed the original forward current or charge, so the energy loss would be minimal.
After the initial charging of CBOOT, D1 only has to replenish the charge lost by supplying gate charge to M1 (via M2) (and possibly some small additional currents consumed by the driver or maybe current-sensing circuits also powered from VBOOT). D1's own capacitance\charge will be negligibly less than that.
If in an IC system, there could be a (very) slight risk using a Schottky -- during the M1,M4 dead time, the switch node is at -0.7 V (or lower because of inductance). This can mean that D1 could charge to more than VLOGIC. If not careful, this could exceed the safe VBOOT..VSWITCH rating.