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.
By the way, the generalisation of $\lambda$ to non-abelian $\mathfrak{g}$ is the following. If $\theta^\alpha$, $e_\alpha$ and $u^\alpha$ are as in the paper, then $$\lambda b = \prod_\alpha (1 - \theta^\alpha i(e_\alpha) )b.$$ This is known as minimal coupling in the Physics literature.