This Fourier transform of $\cos(\sqrt{a^2 - x^2})$ has come up in relation to a physics problem. Searching through Gradshteyn and Rhyzik didn't turn anything up, and trying a number of changes of variable didn't produce anything obvious.
One idea was to make an approximation in which one restricts the range of integration from -a to a, as the contribution outside of this range would be relatively small due to the damping effect of the cosine of an imaginary argument. Then a change of variables with $x = a\sin\phi$ would hopefully lead to a doable integration. This is also not an obvious one. Any help is appreciated!