It looks like a standard lithium cell behind the blue PCB.
If you are running that near 3V, it is permanently DEAD FLAT!
You must keep that battery near 4V for it to have useful capacity; otherwise it is doing nothing.
If the battery was in the correct range it should work fine.
It looks like that PCB may be a battery charge and protection board, but you are not using the correct connections to it & the protected output (that shuts off when the battery is too low) is bypassed?
Get the battery voltage correct and add a 3.3V low drop out voltage regulator to set the supply to the other electronics at 3.3V, if that is what's needed.