Have you considered using AA batteries ?
You can get lithium based AA batteries like Energizer Ultimate Lithium that have something like 3500mAh at around 1.4v..1.7v ... see :
http://data.energizer.com/pdfs/l91.pdfThey're not cheap at around 2-3$ each but it's way cheaper than your 14$ battery.
Regular alkaline AA batteries get close to 2500-2800 mAh and discharge down to around 1..1.2v
You can buy battery holders that solder to the circuit board and then you can pot everything leaving just the battery contacts on the outside. Then insert battery and use hot glue or that selastic / flexible gunk / whatever... something to lock it in the battery holder to prevent vibrations from breaking contact.
There are microcontrollers that run from battery but if you need 3v, why not just use a very efficient voltage doubler IC... would be over 95% efficient at a few mA of current.
Here's some examples:
1. TPS6031x series : 0.9v..1.8v , 2 outputs, 1: 2xVin max 40mA, 2: regulated 3.3v 20mA up to 90% efficiency :
http://www.ti.com/lit/ds/symlink/tps60310.pdf2. TPS6030x series : pretty much same thing?
http://www.ti.com/lit/ds/symlink/tps60300.pdfThe above are ~ 1.7$ in 10pcs or higher and need only 5 ceramic capacitors to work. This + battery is still way cheaper than 14$
There's also very efficient switching regulators.. here's ex
<1$ / 10+ pcs TLV61225 - min 0.7v in, fixed 3.3v out, 94% efficiency, needs only 2 ceramic caps and inductor :
http://www.ti.com/lit/ds/symlink/tlv61225.pdfThere's also TLV61224,
TPS61261 - min 0.8v, fixed 3.3v out or adjustable options, up to 95% efficiency, same min. parts :
http://www.ti.com/lit/ds/symlink/tps61260.pdfEven when you add the price of a surface mounted ceramic inductor and a bunch of ceramic capacitors, you're still below 2$.
There's loads of microcontrollers from Microchip that use less power and can run at lower voltages (like 1.8v..3.6v for example). There's also Silicon Labs with their BusyBee micros that can work at low voltages and they're quite nice micros.
Here's some examples:
https://www.digikey.com/short/pftvj0