I shared the same concern that Ilya expresses in the comments, but after considering it, here is why I think that it works. To have a clean view of the boundary conditions, let's double the disk to the sphere and let everything be equivariant with respect to reflection across the equator.
Moser's theorem truly is a deformation retraction. Let $M$ be a Riemannian manifold, let $\mu$ be a volume form on $M$ (not necessarily Riemannian volume), and let $\phi_\alpha:M \to M$ be a family of diffeomorphisms of $M$ that may or may not preserve $\mu$. Then $\mu_\alpha = (\phi_\alpha)_*(\mu)$ is "wrong". Let $\mu_{\alpha,t}$ be a family of volume forms defined as the weighted geometric mean of $\mu_\alpha$ and $\mu$:$$\mu_{\alpha,t} = \mu^t \mu_\alpha^{1-t}.$$Then there is a corresponding Moser flow $\phi_{\alpha,t}$ such that $\phi_{\alpha,0} = \phi_\alpha$ and $\phi_{\alpha,1}$ is volume-preserving. Moreover, $\phi_{\alpha,t} = \phi_\alpha$ for all $t$ if $\phi_\alpha$ is already volume-preserving for some fixed $\alpha$.
In particular, if $\phi_t$ is a curve of diffeomorphisms as produced by Smale's theorem with $\phi_0$ the identity, then Moser gives you an improvement $\phi_{t,s}$ such that $\phi_{t,1}$ is then what you want. What worried us is whether $\phi_{1,1} = \phi_1$; if $\phi_1$ is area-preserving, then it is true.
