There are plenty of other optimization packages if you want to treat least-square fitting as a general optimization problem.
But if you want to take advantage of the special structure of least-square optimization, some options currently seem to be LsqFit.jl, LeastSquaresOptim.jl, NLLSsolver.jl, and ManOpt.jl. (Update: See replies for more packages.) See also Comparing non-linear least squares solvers
In the particular problem raised in the present thread, however, the lack of robustness was intrinsic to the optimization problem being posed rather than a property of the algorithm, since the minimum was ill-conditioned.