I'm trying to sort out the details of the different notions of image factorization and their interactions.
In the general nonlinear setting, one may define the regular image of an arrow to be the equalizer of its cokernel pair. If the category is moreover pointed, one may also define the kernel of the cokernel. These creatures are isomorphic for pointed sets: the cokernel pair glues along the image and so the equalizer is the image, while the cokernel collapses the image to zero, and so the kernel is the image.
I think they should perhaps be isomorphic in much greater generality, but I'm having trouble getting anywhere with the universal property.
If $\mathsf C$ is finitely bicomplete and pointed, is the kernel of the cokernel isomorphic to the equalizer of the cokernel pair? How to prove this?
You've shown that the kernel of the cokernel is isomorphic the equalizer of the cokernel pair in the category of pointed spaces; but the dual property does not hold! Indeed, the kernel of $f:(A,a_0)\to (B,b_0)$ in the category of pointed spaces is simply $f^{-1}(b_0)$ with its inclusion in $A$, and thus you can show that its cokernel is the quotient map identifying all the elements in $f^{-1}(b_0)$ to a single point, or equivalently, the quotient map for the equivalence relation $x\sim y\Leftrightarrow f(x)=b_0=f(y)$. On the other hand, the kernel pair of $f$ is the usual fibered product $\{(a,a')\in A^2|f(a)=f(a')\}$, with the two projections to $A$, and the coequalizer for this pair of maps is the quotient map for the equivalence relation $x\approx y \Leftrightarrow f(x)=f(y)$. You see that $\sim$ and $\approx$ are not the same equivalence relation, and thus must have different quotient maps; thus the cokernel of the kernel and the coequalizer of the kernel pair are not isomorphic. This means the dual of the category of pointed spaces is a counterexample to the property you suggest.
In fact, it is known (see for example proposition 3.1.23 in Mal'cev, Protomodular, Homological and Semi-Abelian Categories by Francis Borceux and Dominique Bourn, or proposition 2 of $3\times 3$ Lemma and Protomodularity by Dominique Bourn) that regular epimorphisms and cokernels are the same in homological (i.e. pointed regular and protomodular categories). Perhaps surprisingly, the dual of the category of pointed sets is protomodular (and even semi-abelian), but the category of pointed sets is not. This explains why your property works in the category of pointed sets but not in its dual.