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A watch battery?

These are designed to deliver tiny currents for a long time. While your circuit doesn't draw a lot of current it isn't tiny. Worse is that it is pulsed.

You might not think that it is but the EEPROM is pulsing the current. Every time you write a byte it briefly requires more power.

I dug back and found an image of your hardware and I don't see anything that resembles a voltage regulator so the supply voltage is going to change with the changing load. The internal resistance of a CR2032 cell is 10 to 40 Ohms. A 4mA pulsed load is going to put up to a 160mV pulse in the supply. Not good at all.

While the pressure sensor should work just fine over a range of voltages, the voltage must be stable. Changes in the voltage will corrupt the ADC conversion process.

Adding capacitance will reduce the magnitude of the voltage sags but it will not eliminate them. At least for reasonable sized capacitors. You really need a voltage regulator.
 

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