I think that the map $\lambda$ should be defined as follows:
$$\lambda a = a - \theta i(X)a.$$

In this way,
$$D\lambda a = D( a - \theta i(X)a) = da + u i(X)a + \theta di(X)a$$
and using that $di(X) a + i(X) da = 0$ and that $\lambda(i(X) a) = i(X) a$, one finds
$$D\lambda a = \lambda( da + i(X)a u ),$$
which is (4.19) in Atiyah-Bott.