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I am a novice here, but have you looked at Lazarsfeld's chapter "Lectures on linear series" in the book Complex algebraic geometry, IAS Park City math series vol 3? There he deduces it from Reider's theorem which he deduces from Bogomolov's instability theorem, whose proof he also sketches. This is a rewrite of some parts of his chapter on application of vector bundles techniques in the book Lectures on Riemann surfaces from Trieste, where however he says his argument has an error. I am assuming the result you want is that when K is nef then some positive multiple of K is free, (in fact 4K).

yes you are right he does assume general type, but unfortunately for my understanding he does not state that in the theorem itself but only in a paragraph above the theorem, which is then apparently a blanket assumption not restated later. So since you are presumably dealing with the opposite case this is useless to you. On p. 81 of Kollar and Mori they merely state that for surfaces base point freeness is a "non trivial result". This suggests they did not know an easy proof.

edit: In Miles Reid's chapters on algebraic surfaces, he discusses this point explicitly at the end of his treatment of classification of surfaces with K nef. See E.9.1. "Abundance as a logical bottleneck". He states there that he knows only one proof in the literature not using Enriques' argument at a crucial point, namely that in the book of Barth, Peters, and Van de Ven, where they use Ueno's prof of Iitaka's additivity conjecture C(2,1) via moduli of curves. Does that help?