In a well-pointed topos, it is simple to show that $A$ is an initial object if, and only if, there are no global elements $1 \to A$.
Now suppose that $A \to 1$ is an epimorphism. Then $A$ cannot be an initial object since the map $0 \to 1$ is monic and not an isomorphism, hence cannot be epic (since every monic and epic morphism is an isomorphism in a topos). Then $A$ cannot have no global elements, so by the law of excluded middle, $A$ has a global element.
But now suppose the meta-logic here is intuitionistic. That is, on the metatheoretic level, we do not accept the typical classic axioms like $P \lor \neg P$. Can we still prove that $A$ has a global element whenever $A \to 1$ is epi?
If we cannot prove this, what model can we devise of a metatheory together with a well-pointed topos in which this fact is not true?
Note that this is a special case of a more general classical result I am trying to prove, which follows from the special case. The more general theorem is that in a well-pointed topos, a map $f : A \to B$ is epi iff every global element of $B$ factors through $f$.
The fact that for an epi $f : A \to B$, every global element $g : 1 \to B$ factors through $f$ follows by taking the pullback of $g$ and $f$, noting that the resulting map $h : P \to 1$ must be epi since it's the pullback of an epi, and taking a global element of $P$, which gives rise to a global element of $A$ that $g$ factors through.
Note that generally, when we're dealing with well-pointed toposes in ETCS, we add the external Axiom of Choice which states that every epi splits. This immediately resolves the issue. But I am interested in using well-pointed toposes in intuitionistic meta-logic so that the results can apply to intuitionist set-theory and some varieties of intuitionist type-theory, so I would rather avoid imposing any classical axioms at all on either the meta-logic or the internal logic of the topos except well-pointedness.