I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that if $M,N \in C_\lambda$ then there is no elementary embedding from $M$ to $N$ or vice versa (in ZFC)?

One statement of the large cardinal axiom Vopěnka's Principle is that no accessible category has a full subcategory which is both large and discrete.

Adámek and Rosický point out (Remark 6.2(2)) that for any cardinal $\lambda$, it's trivial to come up (in ZFC) with an accessible category with a full discrete subcategory with $\lambda$-many objects. They use the example of the theory $\mathbf{Rel}_\lambda$, with $\lambda$-many unary relation symbols, and the set of objects $A_i$, each carried by the one-point set, where $A_i$ has just the $i$th relation turned "on". Here the accessible category in question is allowed to vary with the cardinal $\lambda$.

But, in ZFC, is there one single accessible category $\mathcal{K}$ which has a full, discrete subcategory $\mathcal{K}_\lambda \subset \mathcal{K}$ of cardinality $\lambda$, for each cardinal $\lambda$?

Of course, the union $\cup_\lambda \mathcal{K}_\lambda$ is large, so (assuming that Vopěnka's principle is consistent over ZFC), if such a category exists, then one won't be able to show that $\cup_\lambda \mathcal{K}_\lambda$ is discrete. But it could be that all of its morphisms go from objects of one $\mathcal{K}_\lambda$ to another $\mathcal{K}_{\lambda'}$, and the $\mathcal{K}_\lambda$'s themselves might all be discrete.

**Bonus question:** in your example, are there clearly morphisms between objects in different $\mathcal{K}_\lambda$'s, or is your example a candidate to become a counterexample to Vopenka in some models (in which connection, this question may be relevant)?