In every-day language, I would say, "Give me the fifth largest pumpkin you have"; that is I would use the ordinal.
However, this feels clunky in mathematical texts, especially when reading out loud: "select the k-th largest element" or, even worse, "the k plus first/oneth largest element".
Is it fair to say/write "the k-largest element"? It seems to me that the "aspect of ordinality" is sufficiently expressed by the -st suffix in larger (or smallest, heaviest, ...), that is the superlative.
$k$th largest
just looks weird, whereas$k$-largest
(do we want/need the hyphen?) works.