It is a theorem of Smale that the group of orientation-preserving diffeomorphisms of $D^2$, rel boundary, is contractible.  If the diffeomorphisms can move the boundary, you can establish a homotopy equivalence between that and the circle.  The diffeomorphisms do not have to preserve area.  Then, a [theorem of Moser][1] establishes a deformation retract from diffeomorphisms to volume-preserving diffeomorphisms.  Moser's result is easier to see if you have a closed manifold, but it extends to manifolds with boundary with the doubling trick.  Together, this indirectly gives you a curve of symplectomorphisms connecting the identity to $\psi$, since in two dimensions the symplectic structure is just a volume structure.  Finally if you have a smooth curve of area-preserving diffeomorphisms of a disk, I think there is a time-dependent Hamiltonian obtained by integrating the corresponding vector field.

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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.

  [1]: https://mathoverflow.net/questions/7817/normal-coordinates-for-a-manifold-with-volume-form