Co-Hausdorffification

The procedure you describe does not produce a $T_2$-space in general. You can read here and here on MO, and also this nice recent bachelor thesis by Bart van Munster. But you do get the Hausdorffification by iterating that procedure (transfinitely many times, in general). It follows that $T_2$-spaces form a reflective (full, replete) subcategory of $\mathbf{Top}$. There is no co-Hausdorffification though, because the only subcategory of $\mathbf{Top}$ that is simultaneously reflective and coreflective is $\mathbf{Top}$ itself: see Kannan, Reflective cum coreflective subcategories in topology, Math. Ann. 195 (1972), 168-174, gdz, and also the 2006 PhD thesis by Martin Sleziak, available here.

Added: Apologies if I misread the question by assuming the factorization to be unique. In order to partially rehabilitate myself let me add some comments. The full subcategory of $T_i$-spaces happens to be reflective for $i=0,1,2,3$ (Kennison characterized reflective subcategories of $\mathbf{Top}$ as those full subcategories which are closed under products and subspaces in $\mathbf{Top}$, see his paper "Reflective functors in general topology and elsewhere", Trans. Amer. Math. Soc. (1965), 303-315). This means that for those values of $i$ there exists a notion of $T_i$-ification (and the factorization is unique). Regarding the weak co-$T_i$-ification (let us call the corresponding weak reflector $WR_i$), at least for $i=1,2,3$ it does not exist by the argument in Eric Wofsey's beautiful answer. If you prefer a published account you can consider Lemma 5.5 in Herrlich's paper "Almost reflective subcategories of $\mathbf{Top}$", Topol. Appl. 49 (1993), 251-264, which states:

Lemma 5.5: If $\alpha$ is an infinite cardinal with successor $\alpha+1$, then there is a continuous map $f\colon 2^{\alpha+1}\to S$ from the $\alpha+1$ power of the discrete 2-point space $2$ onto the Sierpinski space $S$, which does not factor through any $T_1$-space $X$ with $\mathrm{Card}(X)\leq\alpha$.

Then proceed as follows: if such a weak co-$T_i$-ification $WR_i$ existed, take $\alpha$ to be the cardinality of $WR_i(S)$. Observe that $WR_i(S)$ is $T_1$ and $2^{\alpha+1}$ is $T_i$, $i=1,2,3$, being a power of a $T_i$-space, and apply the lemma.

I would be interested to know whether $T_0$-spaces are weakly coreflective in $\mathbf{Top}$.

While johndoe gave the right answer to the question I believe you meant to ask, the question that you did ask is slightly different and also has a negative answer. Specifically, johndoe's answer addresses the version of your question in which the factorization $\bar{f}$ is required to be unique. However, even if you do not require the factorization to be unique, then a co-Hausdorffification still does not exist. (One might call such a thing a weak co-Hausdorffification, by analogy with the term weak limit.)

Specifically, let $X=\{0,1\}$ be the Sierpinski 2-point space (with $1$ closed), and suppose $f:T\to X$ is a map from a Hausdorff space to $X$. Then I claim that there exists a Hausdorff space $Z$ and a map $g:Z\to X$ that does not factor through $f$. To show this, let $\kappa$ be any ordinal of cofinality greater than $|T|$, let $Z=\kappa+1$, and let $g:Z\to X$ send $\kappa$ to $1$ and everything else to $0$. Suppose $h:Z\to T$ is such that $fh=g$. Since $\operatorname{cf}(\kappa)>|T|$, there is an unbounded set $S\subseteq\kappa$ on which $h$ is constant. By continuity of $h$, we must then have $h(\alpha)=h(\kappa)$ for all $\alpha\in S$. But then $g(\alpha)=f(h(\alpha))=f(h(\kappa))=g(\kappa)$ for all $\alpha\in S$, a contradiction.

(In the language of the adjoint functor theorem, this is saying that a co-Hausdorffification fails to exist not only because Hausdorff spaces are not closed under coequalizers, but also because the solution set condition fails. Nevertheless, it turns out that the (large) colimit of all Hausdorff spaces mapping to $X$ does exist in the category of all spaces: it is just $X$ itself. This is not hard to prove; it essentially amounts to showing that any topology is determined by which maps from Hausdorff spaces are continuous.)

Update: As johndoe noted, this argument also rules out a weak co-$T_1$-ification and a weak co-$T_3$-ification, or more generally a weak co-$P$-ification if $P$ is a property which implies $T_1$ and is satisfied by any ordinal under the order topology ($T_1$ is the lower bound because my argument required $T$ to be $T_1$ in order to conclude that $h(\alpha)=h(\kappa)$ for $\alpha\in S$). Here is a slightly different argument that shows there is no weak co-$T_0$-ification:

Let $X=\{0,1\}$ with the indiscrete topology, and suppose $f:T\to X$ is a map from a $T_0$ space to $X$. Then I claim there is a $T_0$ space $Z$ and a map $g:Z\to X$ that does not factor through $f$. As before, let $\kappa$ be an ordinal of cofinality greater than $|T|$; let $Z$ be $\kappa$ equipped with the Alexandrov topology. Let $g:Z\to X$ be the characteristic function of any unbounded and co-unbounded subset of $\kappa$. Suppose $h:Z\to T$ is such that $fh=g$; then as before, $h$ must be constant on an unbounded set. But $h$ must preserve the specialization order, and so since $T$ is $T_0$ this implies that $h$ is eventually constant. Since $g$ is not eventually constant, this is a contradiction.