No. There is no way to change the lsqcurvefit code to use a different measure of error. Ok, no way except for rewriting lsqcurvefit.
The point is, lsqcurvefit uses algorithms that are specific to a sum of SQUARES of residuals. lsqcurvefit is not a general optimizer, that you could somehow just tell it to use a different metric.
If that is your goal, you could in theory use a different tool, perhaps fminunc or some other totally general optimizer. Even that is subject to significant problems however. For example, a sum of absolute values would result in a non-differentiable objective function. That could result in a failure to converge for fminunc.
So IF you truly needed to use a different objective, then you would be best off using an optimizer that would not be subject to such a failure. That might mean fminsearch, or perhaps a genetic algorithm, or some other stochastic scheme like a particle swarm method.
