Still not a full answer, but maybe pointing towards the solution.

    PiecewiseExpand[
     FourierTransform[
      Sinc[\[Omega]1 - \[Omega]2], 
      {\[Omega]1, \[Omega]2}, {t1, t2}, 
      FourierParameters -> {1, -1}
      ],
     t1 \[Element] Reals
     ]

[![enter image description here][1]][1]

This does not work with `Sinc[b (\[Omega]1 - \[Omega]2)]`, though and I don't know why. It should only be a matter of the transformation formula, but apparently, _Mathematica_ is not clever enough to apply it automatically.

  [1]: https://i.sstatic.net/b9WVH.png