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 $ZFC+\{\phi_i\}_{i<n}$, <s>we obtain new cardinals and ordinals in the new universe $V^{ZFC+\{\phi_i\}_{i<n}}$ which in turn yield new surreal numbers in $N_0^{ZFC+\{\phi_i\}_{i<n}}$</s>. (this is wrong, see below)

Has this been studied at all/is it obvious to someone here how to deal with this when trying to study the surreals? 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.

I'm currently 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 we're 'changing' the surreals when we 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.