Generalized semi-metrics taking values on a total order - Which spaces admit one?

Question

I have become interested in metrizability theorems and their generalizations to other notions of a metric. Consider an ordered pair $(M,d)$ and a total order $(T,\leq)$ with least element $0$ where $M$ is a set and $d:M\times M\to T$ is a "generalized semi-metric," i.e. a function satisfying the following axioms:

1. $d(x,y)\geq0$.
2. $d(x,y)=0$ iff $x=y$.
3. $d(x,y)=d(y,x)$.
4. The open balls, sets of the form $B_r(x)=\{y\in M|d(x,y)<r\}$ form a basis for a topology on $M$. Equivalently, the intersection of two non-disjoint open balls contains some other open ball.
5. When $M$ has the topology generated by its open balls, $M\times M$ has the corresponding product topology, and $T$ has the order topology, $d$ is continuous.

A topological space is called "generalized semi-metrizable" if it is generated by a generalized semi-metric. How can we topologically characterize these spaces? I would also be happy with a characterization of spaces satisfying axioms 1-3 or 1-4, although that is not my main interest.

So far I think:

By continuity of $d$, it is necessary that our space $T$ be regular: The proof of Lemma 2.13 in this paper by Stubblefield is given for continuous semi-metrics taking values on $\mathbb{R}$ but seems to generalize to my axioms.

A mistake in the literature claims that a space generated by a continuous semi-metric is actually fully metrizable. This is corrected in Theorem 2.21 of the above paper.

In the case where $T$ has an ordered monoid structure with operation $+$, and $d$ satisfying the triangle inequality, metrizability is well-understood since we have this paper by Ishiu (Theorem 4.5) characterizing these spaces as those which are $\omega_\mu$-metrizable for some ordinal $\mu$, and these have been given topological characterizations. For example, in "Remarks on $\omega_\mu$-additive spaces" by Shu-tang, a space is found to be $\omega_\mu$ metrizable iff it is regular, has an $\aleph_\mu$ locally finite basis (the equivalent of countable locally finite in standard metrizability) and is $\omega_\mu$-additive meaning that the union of strictly less than $\omega_\mu$ closed sets is closed. Other characterizations are found here and here.

What this means is that the class of spaces that are generalized semi-metrizable will fall somewhere between the regular spaces and the regular spaces with $\omega_\mu$-additivity and $\aleph_\mu$ locally finite basis. Is there a more precise characterization that allows us to find which spaces can be generalized semi-metrized? Thank you!