I switched Mosfets for the second go around because I couldn't solder the D2Pak worth a damn.
These are PSMN3R-430-127
As Kiwi says, their on-resistance between 10V and 4.5V gate is almost the same.
I just can't explain why I get so much voltage droop on the boot capacitor even at very small duty cycles.
With those mosfets you should be ok with 3.8v Vgs drive. On resistance will be slightly higher than with 4.5v drive, but less than the difference between 4.5v and 10v. (see fig8 on the data sheet)
I know it is a puzzle why you are not getting the full 5v gate drive. One end of the bootstrap cap is being pulled low by the low side mosfet and the other is pulled to supply voltage thru internal schottky diode, so I would expect the bootstrap caps are actually charging to about 4.5v(assuming 5v supply). Have you measured across them? Rest of the drop is most likely in the drivers but it does seem like a lot. Could they possibly be bad?
Also, how are you measuring the voltages? Are you moving the ground clip on the probe to locations other than ground. If device is powered from a battery shouldn't make that much difference, but would still be better to use two probes in differential mode.