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As Paul Taylor suggests, the proof in the case of $\mathrm{Set}$ seems to rely heavily on epimorphisms splitting. For instance, Linton proves a generalisation of his monadicity theorem for $\mathrm{Set}$-like categories in Theorem 3 of Applied functorial semantics, II, but requires epimorphisms in the base category to split. So I suspect there will not be a similar-looking theorem for presheaf categories.
My understanding is that the list at the start of Sketches is an intuition, not meant as a precise characterisation. Certainly at the time Joyal gave the lecture in 1981, it was not known whether every lex-total category in the sense of Street–Walters is a Grothendieck topos without the size constraint (see Street's Notions of topos).
