I would think that the problem comes from the rank of QRinput being deficient. See for instance MatrixRank, SingularValueList, or Eigenvalues[Transpose[QRinput].QRinput]]. Probably most telling is the bottom left entry of $R$ in QRDecomposition[mat] for each of the matrices QRinput, a, and b. The entry is approximately zero, but nonzero, between $10^{-15}$ and $10^{-14}$. The perturbations of this entry in the matrices are of the same order. These perturbations are of a similar absolute magnitude to the perturbations of the matrices themselves. However, since the value are approximately zero, they are large relative ones. I would expect similarly large relative perturbations in the last row of a matrix $Q$ that satisfies $Q^T R = A$.