Thanks for posting those datasheets. In general, motors tend to draw more power than control boards or small LEDs and it sounds like the coin cells you are trying to use cannot provide enough power to keep the voltage high enough to run the Maestro when the servo is also powered. The second Y axis (IR, ohms and the green line on the graph) in the “Pulse Discharge Characteristics” graph in the lower right hand side of the battery datasheet shows how the internal resistance of the cells varies as the battery discharges. In this case, the IR is about 10 ohms when the battery is fresh and increases up to about 25 ohms just before the useful capacity of the cell is passed. You can think of the internal resistance of a battery as a resistor in series with an ideal cell that will produce a voltage drop from the nominal cell voltage at the output terminals of the battery as current is drawn through the internal resistor. So to use that internal resistance to determine what the voltage of your two coin cells might drop to when trying to supply the 0.5A to 1.5A a small servo like the one on mentioned might require:
First, determine the open circuit voltage of the two cells:
3.0V x 2 = 6.0V
Since each cell has this internal resistance effect in series with it, we have to add the two together to get the total in series resistance:
10ohms + 10 ohms = 20 ohms
To determine the voltage drop across the battery internal resistance when a load much smaller than your motor tries to draw 0.1A through it, we can use Ohm’s law:
0.1A * 20ohms = 2.0V
So we can subtract the voltage drop across the internal resistance from the nominal cell voltage to see what the terminal voltage would be under a small 0.1A load:
6.0V - 2.0V = 4.0V
So even with a small 0.1A load, those two cells cannot deliver enough power to keep their output voltage above the 5V minimum operating voltage of the Maestro.
Generally, we expect the rechargable LiPo batteries commonly available for RC hobbyists to be able to provide more power for a given size than a coin cell. Those types of batteries commonly have a “C” rating (10-20C is common) that you can use with the batteries capacity to see how much current you might expect the battery to be able to deliver (the actual units of the “C” rating are per hour or /h). So if the 300mAh battery you mentioned (300mAh = 0.3Ah) has a 10C rating, you might reasonably expect the battery to be able to provide a peak current of about 3A (0.3Ah * 10/h = 3A). So, to summarize, it might be possible to find a 300mAh LiPo battery that can power your servo.
You might look for a tutorial on Kirchoff’s law and Ohm’s law for more about how to look at current flow though a circuit and how to calculate voltage drop across resistive loads in the circuit.