  This works investigates the Lyapunov-Oseledets spectrum of transfer operator
cocycles associated to one-dimensional random paired tent maps depending on a
parameter $\epsilon$, quantifying the strength of the \emph{leakage} between
two nearly invariant regions. We show that the system exhibits metastability,
and identify the second Lyapunov exponent $\lambda_2^\epsilon$ within an error
of order $\epsilon^2|\log \epsilon|$. This approximation agrees with the naive
prediction provided by a time-dependent two-state Markov chain. Furthermore, it
is shown that $\lambda_1^\epsilon=0$ and $\lambda_2^\epsilon$ are simple, and
are the only exceptional Lyapunov exponents of magnitude greater than $-\log2+
O(\log\log\tfrac 1\epsilon/\log\tfrac 1\epsilon)$.
