There are schematics and links to data sheets for some items. That doesn’t help you but I thought it did bear mentioning.
I don’t know how much of this transfers to the Tiny 2040 but you can power a Pi PICO from a battery via the VSYS pin.
VBUS is the micro-USB input voltage, connected to micro-USB port pin 1. This is nominally 5V (or 0V if the USB is not
connected or not powered).
VSYS is the main system input voltage, which can vary in the allowed range 1.8V to 5.5V, and is used by the on-board
VSYS diode (D1) adds flexibility by allowing power ORing of different supplies into VSYS.
VSYS is the main system ‘input voltage’ and feeds the RT6150 buck-boost SMPS, which generates a fixed 3.3V output
for the RP2040 device and its IO (and can be used to power external circuitry). VSYS is R-C filtered and divided by 3 (by
R5, R6 and C3 in the Pico schematic) and can be monitored on ADC channel 3. This can be used for example as a crude
battery voltage monitor.
The buck-boost SMPS, as its name implies, can seamlessly switch from buck to boost mode, and therefore can
maintain an output voltage of 3.3V from a wide range of input voltages, ~1.8V to 5.5V, which allows a lot of flexibility in
the choice of power source
Raspberry Pi Pico Datasheet