The continuous random energy model (CREM) is a Gaussian process indexed by a binary tree of depth $T$, introduced by Derrida and Spohn, then Bovier and Kurkova, as a toy model of a spin glass. Addario-Berry and the second author have established the existence of an algorithmic hardness threshold $x_*$ in the CREM: finding a state of energy lower than $-x T$ is possible in polynomial time if $x < x_*$, and takes exponential time if $x > x_*$, with high probability. In this article, we are interested in the nature of the transition from polynomial to exponential complexity near the algorithmic hardness threshold. We do this by studying in detail the performance of a certain beam-search algorithm of beam width $N = N(T)$ depending on $T$---we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called $N$-particle branching Brownian motion ($N$-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time $T$ of the time-inhomogeneous $N$-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of $T^{1/3}$. This result appears to be the first of its kind and we believe that this phenomenon should extend to other models.