Understanding the assumptions in the Reverse Fatou's Lemma

For a counter-example to reverse Fatou lemma without the domination hypothesis, take $f_n:=\chi_{(n,n+1)}$, with $X$ the real line, Borel $\sigma$-algebra and Lebesgue measure. We have $\limsup_{n\to +\infty}f_n(x)=0$ for all $x$ but $\int f_nd\mu=1$.

I'd check whether your proof didn't disregard the nonnegativity hypothesis of Fatou's lemma. Those two hypotheses are dual in a sense that's been obfuscated by a bit. If you assumed nonpositivity instead, you'd get a perfect correspondence. Nonpositivity just means being dominated by the zero function so being dominated by an integrable function is a natural generalization.