It seems to me that there is a relatively simple answer to this question but perhaps I am overlooking something.
The category of K-spaces does satisfy the monoid axiom. (If I read the question correctly K-spaces are what is usually called "compactly generated spaces" and compactly generated spaces are what is usually called "weak Hausdorff compactly generated spaces.") We need to check that a (transfinite) sequential colimit of pushouts of maps of the form $X \times i$ where $X$ is an arbitrary K-space and $i$ is an acyclic cofibration is a weak equivalence. First, observe that $X \times i$ is a weak equivalence and while it is not necessarily a Serre cofibration it is a Hurewicz cofibration.
Now, the conclusion will follow if we know the two following facts.
- Pushouts of Hurewicz cofibrations which are weak equivalences are again weak equivalences (and of course Hurewicz cofibrations.)
- (Transfinite) sequential colimits of Hurewicz cofibrations which are weak equivalences are again weak equivalences.
The first one is proven by Boardman and Vogt in Proposition 4.8 (b) in the appendix of Homotopy Invariant Algebraic Structures on Topological Spaces. (The argument doesn't use any fancy point-set topology, in particular separation axioms play no role, so this holds in K-spaces.)
For the second one let's consider a sequence of Hurewicz cofibrations between spaces $(X_\beta \mid \beta < \alpha)$. We want to show that if they are all weak equivalences then so is $X_0 \to \mathrm{colim}_{\beta < \alpha} X_\beta$. First observe that the canonical map from the telescope $\mathrm{Tel}_{\beta < \alpha} \to \mathrm{colim}_{\beta < \alpha} X_\beta$ is a homotopy equivalence so it suffices to show that $X_0 \to \mathrm{Tel}_{\beta < \alpha}$ is a weak equivalence. By fattening the stages of the telescope slightly we can write it as a colimit of open subspaces homotopy equivalent to the original ones. The conclusion follows since compact spaces are small with respect to open embeddings.