However upon connecting the power to the circuit, the voltages on the battery and the output of the regulator drop quite a bit. Is this simply because the battery can't source enough current to the circuit?
Is this simply because the battery can't source enough current to the circuit?
but what about vehicle remote cells?
If they're something like CR2032 then they are rated at less than 1mA, which is a lot less than your micro is demanding. Coin cells are really not suitable as a power source for anything other than memory retention or running RTCs.