A CR2032 battery and a couple of diodes is cheaper.
A battery holder for 2 CR2032 is 3$ and CR2032 batteries are super cheap, like 4 for 1$ or something like that. Add a couple of diodes to drop some voltage (below 5.5v) and you're in business.
If you go with super capacitors, you can get 2.7v rated ones. In the 18650 footprint, you can squeeze 100-120 Farad capacitors.
If you want smaller, you could do for example 47F in 18mm x 33mm :
https://www.digikey.com/product-detail/en/nichicon/JUWT1476MHD/493-4338-ND/2538692You can use a 2.5v LDO to make sure you don't overcharge it, and then use voltage doubler to get up to 2x the voltage out of the supercap ...
See for example the 11 cents MIC5504 :
https://www.digikey.com/product-detail/en/microchip-technology/MIC5504-2.5YM5-TR/576-4880-1-ND/5209408It's a 300mA LDO, with a current limit at around 600mA (min 400mA, typ 630mA, 900mA max) so it would probably be safe to use with that other regulator (if the supercap is fully discharged, you don't want a bigger LDO that would suddenly suck 1-2 A of current to charge the supercap.
Downside is the supercap (if it's big size) will probably take a few minutes to charge up to 2.5v but in your context this isn't an issue.
For voltage doubler, see for example NCP1729 :
https://www.digikey.com/product-detail/en/on-semiconductor/NCP1729SN35T1G/NCP1729SN35T1GOSCT-ND/1967221It does have a current of around 130uA, but you can use the energy of a supercap from 2.5v down to maybe less than 1.5v.
See page 11 of the datasheet for example circuit.
Alternatively, you can buy a 1-2$ step up regulator to 3v or 3.3v or 5v
Here's an example at 2$ :
https://www.ebay.com/itm/3V-3-3V-5V-DC-DC-Boost-Converter-step-up-Voltage-Regulator-Power-Supply-Module/272598733583 They work from 0.7v and up
You could also use one of these regulators with a plain AA or AAA battery. Even non-rechargeable ones would last months in it, at 0.1mA or less current.
You could even go with a crappy NiCD battery and trickle charge it with 1.25v.. 1.45v using a fixed voltage ldo, see for example ablic s817 series
https://www.digikey.com/product-detail/en/ablic-u-s-a-inc/S-817A13APF-CUCTFU/1662-2680-1-ND/7228972https://www.digikey.com/product-detail/en/ablic-u-s-a-inc/S-817B13AMC-CWCT2U/1662-3264-1-ND/8022396or NCP512:
https://www.digikey.com/product-detail/en/on-semiconductor/NCP512SQ13T2G/NCP512SQ13T2GOSCT-ND/2683394