Here is my variation of your basic circuit.
I've moved the power supplies, the comparator, and the gate driver to the left side of the mosfets.
I've swapped the polarity of the comparators inputs. Instead of looking for the C1 voltage to reach some point, it is now looking at the voltage dropped across the inrush limit resistor (R3) to fall, With the values shown the trip point is ~3 Volts. I think that this is better as you're looking at the C3 charge current to drop to a manageable value, instead of watching the C3 voltage to reach a certain level.
There is a spike of current when the S1 closes, but i'ts only about 200pS long, so theres not much effective current into C3 when that ocours. Of course the magnitude and duration will be dependent on parasitic factors of the actual circuit.
Is there something particular about why you chose the LT1713? It seems to me that a generic op-amp or comparator w/pullup would work as well. And using a part that is 12 Volt compliant would eliminate the need for the 5Volt supply.
Also, since this only turns on once, do you really need the gate driver? A npn emitter follower after the op-amp/comparator would probably be enough.