Interior-point methods would be very well suited for this kind of problem. We call it a bound-constrained linear least-squares problem. An easy way to solve it is to rewrite it as
\begin{align*}
\min_{f,r} & \ \|r\|^2 + \alpha \|f\|^2 \\
\mathrm{subject \ to} & \ Kf + r = g \\
& f \geq 0.
\end{align*}
That formulation will avoid any occurrence of K^T K, which could be dense and ill conditioned.
The problem can be solved using RipQP.