Suppose we start in $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^{ZFC}$ denote the surreals in the universe $V^{ZFC}$. If we add some large cardinal assumptions $\{\phi_i\}_{i<\alpha}$ to our axiom list and produce an new theory $ZFC+\{\phi_i\}_{i<n}$, we obtain new cardinals and ordinals in the new universe $V^{ZFC+\{\phi_i\}_{i<n}}$ which in turn yield new surreal numbers in $N_0^{ZFC+\{\phi_i\}_{i<n}}$.

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals algebraically/topologically/categorically? In particular I wonder about new algebraic/topological/categorical/(someday) analytical properties of $N_0^{ZFC+\{\phi_i\}_{i<n}}$ in comparison with $N_0^{ZFC}$, perhaps all viewed from an even larger universe (if necessary for coherence).

For example, I would imagine that $N_0^{ZFC+\text{there exists a supercompact cardinal}}$ has a natural value class "larger" than $N_0^{ZFC}$ since each new inaccessible introduced yields a new possible value.

For my current research I'm worrying about much smaller cardinals (equivalent to Grothendieck universes) for the purpose of letting the Surreals be an object in the category ${\bf CRing}$ -- note here that not only is the class of objects in ${\bf CRing}$ a proper class, some of the objects themselves are also proper classes in an 'uninitiated' universe.

It seems that I'm 'changing' the surreals when I introduce the standard additional large cardinal axioms used in category theory for dealing with proper classes; is this true, and if so how can we understand the changes in a precise manner?

My intuition says that there is an answer along the lines of 'restrict to a certain size surreals and use those', but ideally I would like to work with a version of the surreals containing all known large cardinals to see what they look like, so a more general methodology is desirable.