Fountainhead

As someone said, I would first verify the parasitic draw with nothing attached, put a digital MM between positive lead of battery and the positive battery cable with the amps on meter set to highest scale then work your way down till you find a readable range. Don't use a clamp on meter unless it has range in the 100's of milliamps. Usually quiescent draw is in the low milliamps. This will tell you the quiescent current draw at rest.
Then I would measure the voltage at the battery terminals after the 'lil tyke charger has been connected say, overnight. If the voltage is within tolerance, I would have to assume some other problem exists.
Lets face facts here, you need voltage and amps to crank the car. If the car doesn't crank easily either you have not enough volts or amps, it's that simple.
(Considering your statement that the car cranks normally after a period of time).
If you were feeling REALLY energetic you could measure the current draw of the 'lil tyke battery charger into the battery when first connected, and then after overnight. See if the current draw is over the device rating.