Before getting to wrapped up in costs. it would be a required step to make one work.
can you get your full project to work with a simple MINI ?
once you get it working, you can work on reducing parts counts and pin counts.
if you get an ATTINY85, an 8-pin chip, and a I2C expander and a shift register to deal will all your I/O but spend $1.50 for all the parts, but one microcontroller with 16 pins cost $0.75, you did not save any money?
like wise, if you could use a chip with the correct pin count, but needed to add memory, again, the support chips might cost more than a more expensive chip that does it all.
if you take a NANO or UNO, get all your things working, then figure out how much memory you require, the chips that fit your needs will be much easier to select.
As others have stated, when you have $4 worth of parts on a board, or more like $8 in sensors and $4 in other parts and $5 in enclosure and plugs, The difference between a $0.95 chip and paying $0.75 really do not have much effect in the grand scheme.
there are a lot of options for a microcontroller chip that is $1 USD or less.
figure out the costs of all the other parts, assembly, enclosure, sensors, etc and see if 25 cents savings on a chip has any real effect on your project. There is a real possibility to get a chip for $1. so work with that, figure all the other costs.