Short update:
I've designed a small board which should delay the positive bias until after the negative bias is established. I've milled it, but not yet assembled it. I ended up not using switched capacitors since their output voltage is a bit under the input voltage. The LTC1983-5 would need a Vin between 5.4 and 5.5 V (or so) in order to achieve regulation with 40 mA output (weird).
I'm using an isolated DC-DC converter module instead (though this is also probably a bad idea since I believe it just inverts the voltage, so it'll likely be about as far off since my schematic doesn't have a regulator). I'm going to do some bench testing to see what happens with my circuit later today with some dummy loads.
Some of the parts (for example, the FETs) are just based on what I have available in stock already.
Summary: My new circuit should work with a Vin of about 5V, but provides no regulation, and doesn't provide a way to control the positive bias for gain control. I also don't know what the switching supply module will do if the input voltage drops.