The LM3914 calculator is wrong.
It turns on the 10th LED when the input is 0.95V instead of when its input is the same as its reference voltage which is 1.25V.
It turns on its 1st LED when the input is less than 0.06V but more than 0.05V when it should be 1.25V/10= 0.125V.
It turns on its 2nd LED when its input voltage is a little higher than 1.6V which is not double the voltage that turns on its 1st LED but 1.5V would be correct.
It turns on its 3rd LED when the input voltage is 0.25V when it should be 0.375V, etc.
The LM358 opamp is not good enough for your very high sensitivity because it is noisy (rumble and hiss) and it has a maximum input offset voltage of 7mV.
EDIT: The crossover distortion from the LM358 might cause errors with your measurements.