Dual '4424 drivers
One way around the problem is to use a pair of MIC4424 driver ICs to go between the IR device and the MOSFETs as sketched here.
Normally, the "HO" and the "LO" would go directly to the gates of their respective MOSFETs, but here, the pair of '4424 chips intervene in those signal paths.
Yes, the '4424 is a dual device, but we can only make use of one drive output per chip. We need two '4424 chips because the high-side drive and the low-side drive do not share the same rails.
Just to mention, please notice that the upper FET's series gate resistance is 15 ohms versus the lower FET's series gate resistance of 5.62 ohms. This is deliberately done as one of the measures to avoid "shoot through" where both the upper and lower MOSFET get briefly turned on at the same time and send a HUGE current pulse through the MOSFETs to ground.
The gate drive resistor measure is described by Fairchild Semiconductor here.