Yes, BUT if the problem is ill-conditioned, then computing A'*A is crazy, since that will make it significantly more so. If it cannot succeed because of the ill-conditioning, what does it matter if it is faster?
I would suggest trying an iterative scheme. Perhaps a version of PCCGLS (preconditioned conjugate gradient least squares) There are codes for it in MATLAB. This algorithm does not force you to form A'*A, although you will probably need a preconditioner.