Instead of classes, can't we just use sets of elements of a model? When working in set theory, we often want to work with large collections of sets, so large that they themselves do not form a set. This requires us to have a notion of classes. But then we might want to form a collection of classes, and so on... (For examples, look at the category of all categories, or the collection of gaps of surreal numbers.) This requires stronger and stronger theories (indeed, stronger *languages), if we want to be rigorous.
Instead though, could we just use a model $(M_1, \in_1)$ of set theory? Instead of sets, we would have elements of $M$, and instead of classes, we would have regular sets. If we want to go deeper, we just need a model $(M_2,\in_2) \in M$, and so on...
In particular, if we choose an $(M_2, \in_2)$ such that $(M_1, \in_1) \cong (M_2, \in_2)$, (for example, see Lemma 7 here), then we can form an infinite sequence of models such that $M_1 \ni M_2 \ni_1 M_3 \ni_2 M_4 \ni_3 \dots$ and $(M_1, \in_1) \cong (M_2, \in_2) \cong (M_3, \in_3) \cong (M_4, \in_4) \cong \dots$. We get all this without ever changing our formal system!
So for example, the category of all categories would be a set in $M_1$ containing elements in $c \in M_2$ (corresponding to categories) which would consist of elements of $M_3$ (corresponding to the objects and morphisms of $c$). For example, there would be a $c$ corresponding to the category of all sets(actually the elements of $M_3$, with morphisms being functions in $M_3$).
Due to isomorphism, we would also have a category of all categories in $M_2$, which would be isomorphic to one in $M_1$. It would also be an element of the category of all categories in $M_1$, since the category of all categories is itself a category.
My question is, does this actually work out? Or would issues crop up to stop this construction?
 A: It is an interesting idea. 
Yes, indeed, the countable computably saturated models of set theory are quite remarkable. The purpose of that paper to which you link (A natural model of the multiverse axioms) was to show that this collection of models satisfies all the multiverse axioms that I mentioned in an earlier paper, The set-theoretic multiverse. 
Let me point out a few things.
First, a slightly stronger version of the lemma you mentioned says that every countable computably saturated model of ZFC is isomorphic to some rank-initial segment $V_\beta^M$. So the isomorphic model you find inside is really a very nice model by set-theoretic standards.
Turning this around, and looking at it from the perspective of the smaller model, it follows that every countable computably saturated model is the $V_\beta$ of a taller model, to which it is isomorphic. 
Thus, not only can we make chains of models going down, as you have mentioned, but also we can go up. So we get a chain of models $M_n$ ordered like the integers, which are all isomorphic, and which are all a rank-initial segment $V_\beta$ in the next model. This is something like a chain of Grothendieck universes, but they are all isomorphic. 
It must be mentioned that these kinds of models of set theory, however, are never well-founded, and indeed, the chain you have built shows that $\in^M$ is not well-founded in the model. In particular, the models themselves (which are models of ZFC and think that the ordinals and the $\in$-relation are well-founded), cannot have access to the chain of worlds you have built. 
For example, the isomorphisms are not available inside the models, but only by means of a model-theoretic construction undertaken externally to the models.
