Surreal numbers are the largest possible structure to have a complete order. Games are an extension of the Surreals which only admits a partial order. Along with being larger, smaller or equal to each other, two Games can be fuzzy. For instance, $\left(0|0\right)=*$ is fuzzy to $0$ - they are neither larger, nor smaller nor equal to each other.
Now I wonder: What is left of the Games if you take only those which are countable or maybe computable? Is there a nice construction that more or less directly gives you the countable subset of all Games in a finite number of steps? (Typical constructions I know of would take countably infinitely many steps to construct $\frac{1}{3}$ or any other fraction that isn't of the form $\frac{n}{2^k}$ with $n \in \mathbb{Z}$ and $k \in \mathbb{N}$.)
Unfortunately, division doesn't really make sense when you step outside the surreals. For example, $*n+*n=0$ for all $n$ (e.g. $*+*=0$), so that $0/2$ would no longer be unique. (As an aside, every game has at least one "half" that you can add to itself to get the original game: see this answer.)
For infinite elements and similar things you can use the fact that a copy of the ordinals sits inside the surreals (albeit with non-standard addition and multiplication).
To represent things in a finite way, you can simply use conventional mathematics notation, like $\dfrac{17}\omega-\dfrac{\omega_{1}^{\mathrm{CK}}}{\omega_2}$, and if it's not obvious that can be dealt with in a computable way because math notation isn't linear, just use $\LaTeX$ or similar, like $\texttt{\dfrac{17}\omega-\dfrac{\omega_{1}^{\mathrm{CK}}}{\omega_2}}$. (I included the Church-Kleene ordinal to play with the "computable" part of your question.)