4 of 5 added 52 characters in body

Haskell, 140 bytes

import Data.List
h s|m<-length s=nub[k|l<-head s,k<-("("++l++")"):[u++v|n<-[0..m-2],u<-s!!n,v<-s!!(m-2-n)]]:s
f n=head.(!!n)$iterate h[[""]]

First of all: iterate does create a recursion sequecnce of h, but this is a builtin, and by the rules we may use builtins that use recursion.

This is just a very simple recursive approach, we can generate the n-Dyck-words by putting (n-1)-Dyck-words between paranthesis, or by concatenating from shorter Dyck words. This approach however does generate duplicates, that is why we filter for unique elements using nub.

As Haskell is a functional language, it is almost impossible to do this without recursion. There is an even more direct approach, but that one is banned under the current rules.