Hi Thanura,
Welcome to the forum!!
If space is a concern I’d definitely check out @Liam120347 's topic where he cut down a Pico (a great microcontroller, though getting a battery added to that set up will depend on a few things):
Fitting Pico’s where they probably shouldnt be
While stumbling around the net wasting time I came across this article about how you could cut down a Pico since the bottom porttion didnt have any sensitive components or traces that fed any signals back towards the main processor.
Check out the Tweet that inspired that post: https://twitter.com/PaintYourDragon/status/1376643550536892419?s=20
Warning, you might cut yourself or ruin some Picos if you try this!
I got a Stanley knife and made some deep cuts on the PCB on both sides and snaped the bottom part off a couple.
Now they can fit in much smaller spaces (still the same full size IO, which can be a disadvantage. For $6 this isnt too bad. If you want to spend a bit more but have a PiicoDev/Qwiic/STEMMA port check out some of the QT Py boards Adafruit make
Yeah there are certainly libraries for the SSD1306 for Arduino(its worth having a look around the net for one that suits your project), just note that there arent any official PiicoDev libraries but the underlying hardware should still work!
Back on the topic of batteries, the Vsys pin along with a Schottky diode can accept an input between 5.5 to 1.8V at the pin (3 AA batteries would work well!)
PS: there are some neat things you can do with community contributions to the library such as circles and arcs! GitHub - CoreElectronics/CE-PiicoDev-SSD1306-MicroPython-Module: CE-PiicoDev-SSD1306-MicroPython-Module
Let us know if you have any other questions!
Liam