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Looking for a reference for a version of the "The reverse Lebesgue dominated convergence the...

This is Theorem 1.4 in Bennet and Sharpley's "Interpolation of Operators" (page 3). It actually holds for Banach spaces of functions equipped with what they call "function norms" and not only Orlicz s …
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Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces

That space should be $L_{\exp}$. Check Bennett and Sharpley's "Interpolation of Operators" Chapter 4.6 or look in Rao and Ren's "Theory of Orlicz Spaces".
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