I would love to be using a lithium or LifePo4 battery for this project, the customer wants Alkaline as they are readily available anywhere. The future waste is concerning.
I've desoldered and jumped the AO3402 to find that the MCP16251 sometimes produces 3.3V, sometimes 2V. Capacitors are good, voltage setting resistors are good, inductor is good. Can only determine the IC is bad, and no replacement stock in North America, 45 weeks out for Digi. Feels like a monday
The problem with the first paragraph is that most aa battery chargers charge batteries in pairs, and if your device is gonna use a single aa battery, few people are gonna have a readily available single aa battery that's charged and they won't wait a few hours for a battery to charge, so they're just gonna buy non-rechargeable alkaline and add to the waste pile.
With a lithium battery, the end user no longer needs to replace batteries.
A simple alternative could be to just use 2 AAA batteries instead of a single AA. You get slightly less energy, but user can buy 4 aaa rechargeable, and swap 2 at a time when they're discharged an you get your minimum 2v-ish.
if your device uses little current you may want to check out ratiometric chips, voltage doublers, triplers ... some should handle reverse voltage by design so you could put your mosfet after and use therefore use cheaper mosfet
benefits low quiescent current, high efficiency, low switching noise, no inductor needed just a few ceramic caps and diodes.
Most doublers can be extended to do 3x, so your rechargeable 1.1...1.35v could end up to around 3..3.5v in triple configuration.
See
https://www.digikey.com/short/p28mnpvh