The longest common prefix (LCP) array is a versatile auxiliary data structure in indexed string matching. It can be used to speed up searching using the suffix array (SA) and provides an implicit representation of the topology of an underlying suffix tree. The LCP array of a string of length $n$ can be represented as an array of length $n$ words, or, in the presence of the SA, as a bit vector of $2n$ bits plus asymptotically negligible support data structures. External memory construction algorithms for the LCP array have been proposed, but those proposed so far have a space requirement of $O(n)$ words (i.e. $O(n \log n)$ bits) in external memory. This space requirement is in some practical cases prohibitively expensive. We present an external memory algorithm for constructing the $2n$ bit version of the LCP array which uses $O(n \log σ)$ bits of additional space in external memory when given a (compressed) BWT with alphabet size $σ$ and a sampled inverse suffix array at sampling rate $O(\log n)$. This is often a significant space gain in practice where $σ$ is usually much smaller than $n$ or even constant. We also consider the case of computing succinct LCP arrays for circular strings.