The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < α_*(p) \leq β^*(p) < 1$ such that the length of the longest increasing subsequence in a random permutation of size $n$ sampled from the Brownian separable permuton is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ with probability tending to 1 as $n\to\infty$. In the symmetric case $p=1/2$, we have $α_*(p) \approx 0.812$ and $β^*(p)\approx 0.975$. We present numerical simulations which suggest that the lower bound $α_*(p)$ is close to optimal in the whole range $p\in(0,1)$. Our results work equally well for the closely related Brownian cographons. In this setting, we show that for each $p\in (0,1)$, the size of the largest clique (resp. independent set) in a random graph on $n$ vertices sampled from the Brownian cographon is between $n^{α_*(p) - o(1)}$ and $n^{β^*(p) + o(1)}$ (resp. $n^{α_*(1-p) - o(1)}$ and $n^{β^*(1-p) + o(1)}$) with probability tending to 1 as $n\to\infty$. Our proofs are based on the analysis of a fragmentation process embedded in a Brownian excursion introduced by Bertoin (2002). We expect that our techniques can be extended to prove similar bounds for uniform separable permutations and uniform cographs.