Is it valid to make an admission of a topological space by a "partial quotient map"?
When we implement a space we do not actually implement the space itself, but rather a representation of it. That is, an implementation of $X$ consists of a datatype $T$ and a partial surjection $\delta : T \to X$, as in Type Two Effectivity. When $\delta(p) = x$ we say that the datatype value $p \in T$ represents or realizes the point $x \in X$. Cruicially, there may be many values that represent the same point. Only $T$ exists in the computer, $X$ and $\delta$ are mathematical objects. A particularly well-behaved kind of representation is an admissible one, as you mention.
In your case, you correctly implemented an admissible representation of the Sierpinski space: $X$ is the Sierpinski space, $T$ is Sierpinski, and the surjection $\delta$ maps $\omega$ to $\bot$ and $\mathtt{Succ}^n \,\mathtt{Zero}$ to $\top$. There is no need for further quotienting.
The attempt to define == on Sierpinski is not about quotienting, but rather trying to define the characteristic map $\mathrm{eq} : X \to \{0,1\}$ of equality as a map into Booleans, which is only possible when the space is discrete. You discovered that the Sierpinski space is not discrete.
If we have a representation $\delta : T \to X$ then we can form the representation of a quotient $Y = X/{\sim}$ as $\delta' : T \to Y$ where $\delta'(p) = [\delta(p)]_{\sim}$. That is, quotienting affects the map but not the datatype.