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Testing continued by removing 1 of the capacitors after the FETs to bring the capacitance down to 100uF total. This seemed to do the trick, but is this amount of capacitance required? With only 1 capacitor on the board, I could not get it to trigger the reset loop condition.
Solutions to Discuss
Letting the capacitors discharge seems to be a counter-intuitive way to get the inrush current low enough that the effects of resistance do not cause a substantial voltage drop - you would think that discharging more would give a larger inrush current - so I went searching in the datasheet for soft-start since I seem to remember seeing that somewhere
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