I still don't think that the relative amount of rotation of the Earth compared to that of the Moon is enough to induce the amount of angular momentum needed to increase it's orbit by 3cm a year.
You think wrongly.
The Earth slows its rotation due to interaction with the Moon by some 1.5 milliseconds per century. It is this angular momentum lost by the Earth that is given to to the Moon.
The total kinetic energy of a rotating sphere can be found by
$$E = \frac{\omega^2 M r^2}{5}$$
Where w is the angular velocity in radian/sec.
M is its mass
r is its radius
Plugging in the values for the Earth for both its present rate of rotation and the amount it would have slowed in one year, and taking the difference, we get a value of:
4.34E+18 joules.
This is the amount of energy the Earth has to transfer to the Moon.
The total energy of the orbiting Moon is found by
$$E= \frac{GMmMe}{2a}$$
Where G is the gravitational constant
Me and Mm are the masses of the Earth and the Moon
a is the average orbital radius for the Moon.
Plugging in the correct values for the Moon's present orbit and one 3.8 cm further out and taking the difference we get a value of:
3.77E+18 joules
Which is the amount of energy it would take to raise the Moon's orbit by 3.8 cm and is
less than the amount of energy the Earth has to transfer in one year.
The difference is due to the fact that some of the energy given up by the Earth is lost due to tidal heating.
So yes, the Earth does give up enough rotational energy to the Moon in order to increase its orbit by 3.8 cm per year.