= Solution
\b[No infinite cardinal is categorical for this <first-order theory>[theory]], as described by <no categorical cardinal for the complete named rational diagram>. For an arbitrary infinite $\kappa$, choose a dense linear order $I_\kappa$ without endpoints of size $\kappa$; for example the <lexicographic order> on $\kappa\times\mathbb Q$ works, since every rational block is dense and has no endpoints.
Construct one order by inserting $I_\kappa$ into a fixed irrational cut of the named rational order. Every element of this order is bounded above by some named rational. Construct a second order by putting an $I_\kappa$ tail above all named rationals, as the ordered sum $\mathbb Q+I_\kappa$. Both are dense linear orders without endpoints, both have <cardinality> $\kappa$, and both contain the named rational copy as an <elementary substructure> by <quantifier elimination>. Hence both satisfy its <elementary diagram of a structure>[complete diagram] $T$.
They are not isomorphic over the named constants: the second has an element greater than every named rational, and the first has none. This property need not be expressible by a single sentence in the infinite language to be an <isomorphism of structures>[isomorphism] invariant. Finite cardinalities have no <first-order model>[models] at all because the constants name infinitely many distinct <Elements>, so the existence clause excludes categoricity there too.
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