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I noticed you are using fixed point iteration, which doesn't have the quickest convergence rate (or may not converge at all if $\lVert \hat{\gamma}\rVert > m$). You can speed this up using a quasi-Newton method. See Woodbury's matrix identity.
I looked into solving outrigger systems, and for my particular equation it would run in $O(n^3)$ time. It might be fast enough, but I would prefer an $O(n^2\log n)$ algorithm if I can find one.
@MaximUmansky that looks promising, but eq. (2) has diagonals that aren't all in one band. This answer has more info on solving this specific problem. I'll look into it.
