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.